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Difference between revisions of "2018 AMC 10A Problems/Problem 23"

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(→‎Video Solution (#21-#25))
 
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−
{{duplicate|[[2018 AMC 12A Problems|2018 AMC 12A #17]] and [[2018 AMC 10A Problems|2018 AMC 10A #23]]}}
+
{{duplicate|[[2018 AMC 10A Problems/Problem 23|2018 AMC 10A #23]] and [[2018 AMC 12A Problems/Problem 17|2018 AMC 12A #17]]}}
  
 
== Problem ==
 
== Problem ==
  
−
Farmer Pythagoras has a field in the shape of a right triangle. The right triangle's legs have lengths 3 and 4 units. In the corner where those sides meet at a right angle, he leaves a small unplanted square <math>S</math> so that from the air it looks like the right angle symbol. The rest of the field is planted. The shortest distance from <math>S</math> to the hypotenuse is 2 units. What fraction of the field is planted?
+
Farmer Pythagoras has a field in the shape of a right triangle. The right triangle's legs have lengths <math>3</math> and <math>4</math> units. In the corner where those sides meet at a right angle, he leaves a small unplanted square <math>S</math> so that from the air it looks like the right angle symbol. The rest of the field is planted. The shortest distance from <math>S</math> to the hypotenuse is <math>2</math> units. What fraction of the field is planted?
  
 
<asy>
 
<asy>
−
draw((0,0)--(4,0)--(0,3)--(0,0));
+
/* Edited by MRENTHUSIASM */
−
draw((0,0)--(0.3,0)--(0.3,0.3)--(0,0.3)--(0,0));
+
size(160);
−
fill(origin--(0.3,0)--(0.3,0.3)--(0,0.3)--cycle, gray);
+
pair A, B, C, D, F;
−
label("$4$", (2,0), N);
+
A = origin;
−
label("$3$", (0,1.5), E);
+
B = (4,0);
−
label("$2$", (.8,1), E);
+
C = (0,3);
−
label("$S$", (0,0), NE);
+
D = (2/7,2/7);
−
draw((0.3,0.3)--(1.4,1.9), dashed);
+
F = foot(D,B,C);
 +
fill(A--(2/7,0)--D--(0,2/7)--cycle, lightgray);
 +
draw(A--B--C--cycle);
 +
draw((2/7,0)--D--(0,2/7));
 +
label("$4$", midpoint(A--B), N);
 +
label("$3$", midpoint(A--C), E);
 +
label("$2$", midpoint(D--F), SE);
 +
label("$S$", midpoint(A--D));
 +
draw(D--F, dashed);
 
</asy>
 
</asy>
  
 
<math>\textbf{(A) }  \frac{25}{27}  \qquad        \textbf{(B) }  \frac{26}{27}  \qquad    \textbf{(C) }  \frac{73}{75}  \qquad  \textbf{(D) } \frac{145}{147} \qquad  \textbf{(E) }  \frac{74}{75} </math>
 
<math>\textbf{(A) }  \frac{25}{27}  \qquad        \textbf{(B) }  \frac{26}{27}  \qquad    \textbf{(C) }  \frac{73}{75}  \qquad  \textbf{(D) } \frac{145}{147} \qquad  \textbf{(E) }  \frac{74}{75} </math>
  
−
==Solution 1==
+
==Solution 1 (Area Addition)==
 +
Note that the hypotenuse of the field is <math>5,</math> and the area of the field is <math>6.</math> Let <math>x</math> be the side-length of square <math>S.</math>
 +
 
 +
We partition the field into a red triangle, a yellow triangle, and a green triangle, as shown below:
 +
<asy>
 +
/* Edited by MRENTHUSIASM */
 +
size(180);
 +
pair A, B, C, D, F;
 +
A = origin;
 +
B = (4,0);
 +
C = (0,3);
 +
D = (2/7,2/7);
 +
F = foot(D,B,C);
 +
fill(A--D--C--cycle, red);
 +
fill(A--D--B--cycle, yellow);
 +
fill(B--D--C--cycle, green);
 +
draw(A--B--C--cycle);
 +
label("$5$", midpoint(B--C), NE);
 +
label("$4$", midpoint(A--B), S);
 +
label("$3$", midpoint(A--C), W);
 +
label("$2$", midpoint(D--F), SE);
 +
label("$S$", midpoint(A--D));
 +
label("$x$", midpoint((0,2/7)--D), N);
 +
label("$x$", midpoint((2/7,0)--D), E);
 +
draw((2/7,0)--D--(0,2/7));
 +
draw(A--D^^B--D^^C--D, dashed);
 +
draw(D--F, dashed);
 +
</asy>
 +
Let the brackets denote areas. By area addition, we set up an equation for <math>x:</math>
 +
<cmath>\begin{align*}
 +
[\text{Red Triangle}]+[\text{Yellow Triangle}]+[\text{Green Triangle}]&=[\text{Field}] \\
 +
\frac{3x}{2}+\frac{4x}{2}+\frac{5\cdot2}{2}&=6,
 +
\end{align*}</cmath>
 +
from which <math>x=\frac27.</math> Therefore, the answer is <cmath>\frac{[\text{Field}]-[S]}{[\text{Field}]}=\frac{6-x^2}{6}=\boxed{\textbf{(D) } \frac{145}{147}}.</cmath>
 +
~MRENTHUSIASM
 +
 
 +
==Solution 2 (Area Addition)==
 
Let the square have side length <math>x</math>. Connect the upper-right vertex of square <math>S</math> with the two vertices of the triangle's hypotenuse. This divides the triangle in several regions whose areas must add up to the area of the whole triangle, which is <math>6</math>.
 
Let the square have side length <math>x</math>. Connect the upper-right vertex of square <math>S</math> with the two vertices of the triangle's hypotenuse. This divides the triangle in several regions whose areas must add up to the area of the whole triangle, which is <math>6</math>.
−
 
 
<asy>
 
<asy>
−
draw((0,0)--(4,0)--(0,3)--(0,0));
+
/* Edited by MRENTHUSIASM */
−
draw((0,0)--(0.3,0)--(0.3,0.3)--(0,0.3)--(0,0));
+
size(180);
−
fill(origin--(0.3,0)--(0.3,0.3)--(0,0.3)--cycle, gray);
+
pair A, B, C, D, F;
−
label("$4$", (2,0), S);
+
A = origin;
−
label("$3$", (0,1.5), W);
+
B = (4,0);
−
label("$2$", (.8,1), E);
+
C = (0,3);
−
label("$S$", (0,0), NE);
+
D = (2/7,2/7);
−
draw((0.3,0.3)--(1.4,1.9), dashed);
+
F = foot(D,B,C);
−
draw((0.3,0.3)--(4,0), dashed);
+
fill(A--(2/7,0)--D--(0,2/7)--cycle, lightgray);
−
draw((0.3,0.3)--(0,3), dashed);
+
draw(A--B--C--cycle);
−
label("$\small{x}$", (0.15,0.3), N);
+
label("$5$", midpoint(B--C), NE);
−
label("$\small{x}$", (0.3,0.15), E);
+
label("$4$", midpoint(A--B), S);
 +
label("$3$", midpoint(A--C), W);
 +
label("$2$", midpoint(D--F), SE);
 +
label("$S$", midpoint(A--D));
 +
label("$x$", midpoint((0,2/7)--D), N);
 +
label("$x$", midpoint((2/7,0)--D), E);
 +
draw((2/7,0)--D--(0,2/7));
 +
draw(B--D^^C--D, dashed);
 +
draw(D--F, dashed);
 
</asy>
 
</asy>
 +
Square <math>S</math> has area <math>x^2</math>, and the two thin triangle regions have area <math>\dfrac{x(3-x)}{2}</math> and <math>\dfrac{x(4-x)}{2}</math>. The final triangular region with the hypotenuse as its base and height <math>2</math> has area <math>5</math>. Thus, we have <cmath>x^2+\dfrac{x(3-x)}{2}+\dfrac{x(4-x)}{2}+5=6.</cmath>
 +
Solving gives <math>x=\dfrac{2}{7}</math>. The area of <math>S</math> is <math>\dfrac{4}{49}</math> and the desired ratio is <math>\dfrac{6-\tfrac{4}{49}}{6}=\boxed{\textbf{(D) } \frac{145}{147}}</math>.
  
−
Square <math>S</math> has area <math>x^2</math>, and the two thin triangle regions have area <math>\dfrac{x(3-x)}{2}</math> and <math>\dfrac{x(4-x)}{2}</math>. The final triangular region with the hypotenuse as its base and height <math>2</math> has area <math>5</math>. Thus, we have <cmath>x^2+\dfrac{x(3-x)}{2}+\dfrac{x(4-x)}{2}+5=6</cmath>
+
Alternatively, once you get <math>x=\frac{2}{7}</math>, you can avoid computation by noticing that there is a denominator of <math>7</math>, so the answer must have a factor of <math>7</math> in the denominator, which only <math>\frac{145}{147}</math> does.
  
−
Solving gives <math>x=\dfrac{2}{7}</math>. The area of <math>S</math> is <math>\dfrac{4}{49}</math> and the desired ratio is <math>\dfrac{6-\dfrac{4}{49}}{6}=\boxed{\dfrac{145}{147}}</math>.
+
==Solution 3 (Similar Triangles)==
 +
Let the square have side length <math>s</math>. If we were to extend the sides of the square further into the triangle until they intersect on point on the hypotenuse, we'd have a similar right triangle formed between the hypotenuse and the two new lines, and two smaller similar triangles that share a side of length <math>2</math>. Using the side-to-side ratios of these triangles, we can find that the length of the larger similar triangle is <math>\frac{5}{3}(2)=\frac{10}{3}</math>. Now, let's extend this larger similar right triangle to the left until it hits the side of length <math>3</math>. Now, the length is <math>\frac{10}{3}+s</math>, and using the ratios of the side lengths, the height is <math>\frac{3}{4}\left(\frac{10}{3}+s\right)=\frac{5}{2}+\frac{3s}{4}</math>. Looking at the diagram, if we add the height of this triangle to the side length of the square, we'd get <math>3</math>, so
 +
<cmath>\begin{align*}
 +
\frac{5}{2}+\frac{3s}{4}+s&=3 \\
 +
\frac{5}{2}+\frac{7s}{4}&=3 \\
 +
\frac{7s}{4}&=\frac{1}{2} \\
 +
s&=\frac{2}{7}.
 +
\end{align*}</cmath>
 +
So, the area of the square is <math>\left(\frac{2}{7}\right)^2=\frac{4}{49}</math>.
  
−
Alternatively, once you get <math>x=\frac{2}{7}</math>, you can avoid computation by noticing that there is a denominator of <math>7</math>, so the answer must have a factor of <math>7</math> in the denominator, which only <math>\boxed{\dfrac{145}{147}}</math> does.
+
Now comes the easy part--finding the ratio of the areas: <math>\frac{3\cdot 4 \cdot \frac{1}{2} -\frac{4}{49}}{3\cdot 4 \cdot \frac{1}{2}}=\frac{6-\frac{4}{49}}{6}=\frac{294-4}{294}=\frac{290}{294}=\boxed{\textbf{(D) } \frac{145}{147}}</math>.
  
−
==Solution 2==
+
==Solution 4 (Similar Triangles)==
−
Let the square have side length <math>s</math>. If we were to extend the sides of the square further into the triangle until they intersect on point on the hypotenuse, we'd have a similar right triangle formed between the hypotenuse and the two new lines, and 2 smaller similar triangles that share a side of length 2. Using the side-to-side ratios of these triangles, we can find that the length of the larger similar triangle is <math>\frac{5}{3}(2)=\frac{10}{3}</math>. Now, let's extend this larger similar right triangle to the left until it hits the side of length 3. Now, the length is <math>\frac{10}{3}+s</math>, and using the ratios of the side lengths, the height is <math>\frac{3}{4}(\frac{10}{3}+s)=\frac{5}{2}+\frac{3s}{4}</math>. Looking at the diagram, if we add the height of this triangle to the side length of the square, we'd get 3, so <cmath>\frac{5}{2}+\frac{3s}{4}+s=\frac{5}{2}+\frac{7s}{4}=3 \\ \frac{7s}{4}=\frac{1}{2} \\ s=\frac{2}{7} \implies \textrm{ area of square is } (\frac{2}{7})^2=\frac{4}{49}</cmath>
+
<asy>
 +
/* Edited by MRENTHUSIASM */
 +
size(180);
 +
pair A, B, C, D, F;
 +
A = origin;
 +
B = (4,0);
 +
C = (0,3);
 +
D = (2/7,2/7);
 +
F = foot(D,B,C);
 +
fill(A--(2/7,0)--D--(0,2/7)--cycle, lightgray);
 +
draw(A--B--C--cycle);
 +
label("$4$", midpoint(A--B), S);
 +
label("$3$", midpoint(A--C), W);
 +
label("$2$", midpoint(D--F), SE);
 +
label("$S$", midpoint(A--D));
 +
label("$\ell$", midpoint((0,2/7)--D), N);
 +
label("$\ell$", midpoint((2/7,0)--D), E);
 +
label("$\ell$", midpoint((2/7,2/7+5/2)--(0,2/7+5/2)), S);
 +
label("$\ell$", midpoint((2/7+10/3,2/7)--(2/7+10/3,0)), W);
 +
label("$\frac{5}{2}$", midpoint((2/7,2/7+5/2)--D), E);
 +
label("$\frac{10}{3}$", midpoint((2/7+10/3,2/7)--D), N);
 +
draw((2/7,0)--D--(0,2/7));
 +
draw((2/7,2/7+5/2)--D^^(2/7+10/3,2/7)--D, dashed);
 +
draw((2/7,2/7+5/2)--(0,2/7+5/2)^^(2/7+10/3,2/7)--(2/7+10/3,0), dashed);
 +
draw(D--F, dashed);
 +
</asy>
 +
On the diagram above, find two smaller triangles similar to the large one with side lengths <math>3</math>, <math>4</math>, and <math>5</math>; consequently, the segments with length <math>\frac{5}{2}</math> and <math>\frac{10}{3}</math>.
  
−
Now comes the easy part: finding the ratio of the areas: <math>\frac{3\cdot 4 \cdot \frac{1}{2} -\frac{4}{49}}{3\cdot 4 \cdot \frac{1}{2}}=\frac{6-\frac{4}{49}}{6}=\frac{294-4}{294}=\frac{290}{294}=\boxed{\frac{145}{147}}</math>.
+
With <math>\ell</math> being the side length of the square, we need to find an expression for <math>\ell</math>. Using the hypotenuse, we can see that <math>\frac{3}{2}+\frac{8}{3}+\frac{5}{4}\ell+\frac{5}{3}\ell=5</math>. Simplifying, <math>\frac{35}{12}\ell=\frac{5}{6}</math>, or <math>\ell=\frac27</math>.
  
−
==Solution 3==
+
A different calculation would yield <math>\ell+\frac{3}{4}\ell+\frac{5}{2}=3</math>, so <math>\frac{7}{4}\ell=\frac{1}{2}</math>. In other words, <math>\ell=\frac{2}{7}</math>, while to check, <math>\ell+\frac{4}{3}\ell+\frac{10}{3}=4</math>. As such, <math>\frac{7}{3}\ell=\frac{2}{3}</math>, and <math>\ell=\frac{2}{7}</math>.
−
We use coordinate geometry. Let the right angle be at <math>(0,0)</math> and the hypotenuse be the line <math>3x+4y = 12</math> for <math>0\le x\le 3</math>. Denote the position of <math>S</math> as <math>(s,s)</math>, and by the point to line distance formula, we know that <cmath>\frac{|3s+4s-12|}{5} = 2</cmath> <cmath>\Rightarrow |7s-12| = 10</cmath> Obviously <math>s<\frac{22}{7}</math>, so <math>s = \frac{2}{7}</math>, and from here the rest of the solution follows to get <math>\boxed{\frac{145}{147}}</math>.
 
  
−
==Solution 4==
+
Finally, we get <math>A(\Square S)=\ell^2=\frac{4}{49}</math>, to finish. As a proportion of the triangle with area <math>6</math>, the answer would be <math>1-\frac{4}{49\cdot6}=1-\frac{2}{147}=\frac{145}{147}</math>, so <math>\boxed{\textbf{(D) } \frac{145}{147}}</math> is correct.
 +
 
 +
==Solution 5 (Similar Triangles)==
 
Let the side length of the square be <math>x</math>. First off, let us make a similar triangle with the segment of length <math>2</math> and the top-right corner of <math>S</math>. Therefore, the longest side of the smaller triangle must be <math>2 \cdot \frac54 = \frac52</math>. We then do operations with that side in terms of <math>x</math>. We subtract <math>x</math> from the bottom, and <math>\frac{3x}{4}</math> from the top. That gives us the equation of <math>3-\frac{7x}{4} = \frac{5}{2}</math>. Solving, <cmath>12-7x = 10 \implies x = \frac{2}{7}.</cmath>
 
Let the side length of the square be <math>x</math>. First off, let us make a similar triangle with the segment of length <math>2</math> and the top-right corner of <math>S</math>. Therefore, the longest side of the smaller triangle must be <math>2 \cdot \frac54 = \frac52</math>. We then do operations with that side in terms of <math>x</math>. We subtract <math>x</math> from the bottom, and <math>\frac{3x}{4}</math> from the top. That gives us the equation of <math>3-\frac{7x}{4} = \frac{5}{2}</math>. Solving, <cmath>12-7x = 10 \implies x = \frac{2}{7}.</cmath>
 +
Thus, <math>x^2 = \frac{4}{49}</math>, so the fraction of the triangle (area <math>6</math>) covered by the square is <math>\frac{2}{147}</math>. The answer is then <math>\boxed{\textbf{(D) } \frac{145}{147}}</math>.
  
−
Thus, <math>x^2 = \frac{4}{49}</math>, so the fraction of the triangle (area <math>6</math>) covered by the square is <math>\frac{2}{147}</math>. The answer is then <math>\boxed{\dfrac{145}{147}}</math>.
+
==Solution 6 (Similar Triangles)==
 +
<asy>
 +
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */
 +
import graph; size(180);
 +
real labelscalefactor = 1.5; /* changes label-to-point distance */
 +
// pen dps = linewidth(0.5) + fontsize(10);
 +
// defaultpen(dps); /* default pen style */
 +
// pen dotstyle = black; /* point style */  
 +
real xmin = -1.6030465381283199, xmax = 7.095084767820557, ymin = -1.3624649422453508, ymax = 4.065350676871526;  /* image dimensions */
  
−
==Solution 5==
+
/* draw figures */
−
<asy>
+
draw((0,0)--(0,3));  
−
draw((0,0)--(4,0)--(0,3)--(0,0));
+
draw((0,0)--(4,0));  
−
draw((0,0)--(0.3,0)--(0.3,0.3)--(0,0.3)--(0,0));
+
draw((4,0)--(0,3));  
−
fill(origin--(0.3,0)--(0.3,0.3)--(0,0.3)--cycle, gray);
+
draw((0,0.2857142857142857)--(0.2857142857142857,0.2857142857142857));  
−
draw((0.3,0.3)--(3.6,0.3), dashed);
+
draw((0.2857142857142857,0.2857142857142857)--(0.2857142857142857,0));  
−
draw((0.3,2.7)--(0.3,0.3), dashed);
+
draw((0.07142857142857142,0)--(1.4857142857142858,1.885714285714286));  
−
label("$S$", (-0.05,-0.05), NE);
+
label("$A$",(0, 0),SW*labelscalefactor);  
−
draw((0.3,0.3)--(1.41,1.91));
+
label("$B$",(4,0),SE*labelscalefactor);  
−
draw((1.63,1.78)--(1.48,1.56));
+
label("$C$",(0, 3),N*labelscalefactor);  
−
draw((1.28,1.70)--(1.48,1.56));
+
label("$D$",(0.2857142857142857,0),S*labelscalefactor);  
−
label("$4$", (2,0), S);
+
label("$E$",(0.2857142857142857, 0.2857142857142857),E*labelscalefactor);  
−
label("$3$", (0,1.5), W);
+
label("$F$",(0.0714285714, 0),S*labelscalefactor);  
−
label("$\frac{10}{3}$", (2,0.3), N);
+
label("$G$", (1.49, 1.89), NE*labelscalefactor);
−
label("$\frac{5}{2}$", (0.3,1.5), E);
+
/* dots and labels */
−
label("$2$", (1,1.2), E);
+
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);  
−
draw((3.6,0)--(3.6,0.3), dashed);
+
/* end of picture */
−
draw((0,2.7)--(0.3,2.7), dashed);
 
−
label("$\small{l}$", (3.6,0.15), W);
 
−
label("$\small{l}$", (0.15,2.7), S);
 
−
label("$\small{l}$", (0.3,0.15), E);
 
−
label("$\small{l}$", (0.15,0.3), N);
 
 
</asy>
 
</asy>
 +
Let <math>AD=x</math>. Note that <math>\triangle DEF</math> is a <math>3{-}4{-}5</math> triangle, so <math>EF=\frac{5}{4}x</math> and <math>FD=\frac{3}{4}x</math>. <math>BF=BD+FD=4-x+\frac{3}{4}x=4-\frac{1}{4}x</math>. We know that <math>GE</math> is <math>2</math> from the problem so <math>GF=2+\frac{5}{4}x</math>. <math>\triangle FGB</math> is also a <math>3{-}4{-}5</math> triangle with <math>GF:BF=3:5</math>. We now have <math>3\left(4-\frac{1}{4}x\right)=5\left(2+\frac{5}{4}x\right)</math>. Solving this equation, we get that <math>x=\frac{2}{7}</math> so the area of <math>S</math> is <math>\frac{4}{49}</math>. The area of the triangle is <math>\frac{3\cdot 4}{2}=6</math> so the fraction of field that is unplanted is <math>\frac{\frac{4}{49}}{6}=\frac{2}{147}</math>. Thus, the fraction of the field that is planted is <math>1-\frac{2}{147}=\boxed{\textbf{(D) } \frac{145}{147}}</math>.
  
−
On the diagram above, find two smaller triangles similar to the large one with side lengths <math>3</math>, <math>4</math>, and <math>5</math>; consequently, the segments with length <math>\frac{5}{2}</math> and <math>\frac{10}{3}</math>.
+
~Heavytoothpaste
  
−
With <math>l</math> being the side length of the square, we need to find an expression for <math>l</math>. Using the hypotenuse, we can see that <math>\frac{3}{2}+\frac{8}{3}+\frac{5}{4}l+\frac{5}{3}l=5</math>. Simplifying, <math>\frac{35}{12}l=\frac{5}{6}</math>, or <math>l=2/7</math>.
+
==Solution 7 (Coordinate Geometry)==
 +
We use coordinate geometry. Let the right angle be at <math>(0,0)</math> and the hypotenuse be the line <math>3x+4y = 12</math> for <math>0\le x\le 3</math>. Denote the position of <math>S</math> as <math>(s,s)</math>, and by the point to line distance formula, we know that
 +
<cmath>\begin{align*}
 +
\frac{|3s+4s-12|}{5} &= 2 \\
 +
|7s-12| &= 10
 +
\end{align*}</cmath>
 +
Solving this, we get <math>s=\frac{22}{7}, \frac{2}{7}</math>. Obviously <math>s<\frac{22}{7}</math>, so <math>s = \frac{2}{7}</math>, and from here, the rest of the solution follows to get <math>\boxed{\textbf{(D) } \frac{145}{147}}</math>.
  
−
A different calculation would yield <math>l+\frac{3}{4}l+\frac{5}{2}=3</math>, so <math>\frac{7}{4}l=\frac{1}{2}</math>. In other words, <math>l=\frac{2}{7}</math>, while to check, <math>l+\frac{4}{3}l+\frac{10}{3}=4</math>. As such, <math>\frac{7}{3}l=\frac{2}{3}</math>, and <math>l=\frac{2}{7}</math>.
+
==Solution 8 (Coordinate Geometry)==
 +
Let the right angle be at <math>(0,0)</math>, the point <math>(x,x)</math> be the far edge of the unplanted square and the hypotenuse be the line <math>y=-\frac{3}{4}x+3</math>. Since the line from <math>(x,x)</math> to the hypotenuse is the shortest possible distance, we know this line, call it line <math>\l</math>, is perpendicular to the hypotenuse and therefore has a slope of <math>\frac{4}{3}</math>.  
  
−
Finally, we get <math>A(\Square S)=l^2=\frac{4}{49}</math>, to finish. As a proportion of the triangle with area <math>6</math>, the answer would be <math>1-\frac{4}{49\cdot6}=1-\frac{2}{147}=\frac{145}{147}</math>, so <math>\boxed{\textit D}</math> is correct.
+
Since we know <math>m=\frac{4}{3}</math> , we can see that the line rises by <math>\frac{8}{5}</math> and moves to the right by <math>\frac{6}{5}</math> to meet the hypotenuse. (Let <math>2 = 5x</math> and the rise be <math>4x</math> and the run be <math>3x</math> and then solve.) Therefore, line <math>\l</math> intersects the hypotenuse at the point <math>\left(x+\frac{6}{5}, x+\frac{8}{5}\right)</math>. Plugging into the equation for the hypotenuse we have <math>x=\frac{2}{7}</math> , and after a bit of computation we get <math>\boxed{\textbf{(D) } \frac{145}{147}}</math>.
  
−
== Solution 6: Also Coordinate Geo ==
+
==Solution 9 (Pythagorean Theorem)==
−
Let the right angle be at <math>(0,0)</math>, the point <math>(x,x)</math> be the far edge of the unplanted square and the hypotenuse be the line <math>y=-\frac{3}{4}x+3</math>. Since the line from <math>(x,x)</math> to the hypotenuse is the shortest possible distance, we know this line, call it line <math>\l</math>, is perpendicular to the hypotenuse and therefore has a slope of <math>\frac{4}{3}</math>.  
+
Let the side length of the square be <math>a</math>, and the lengths that the line from <math>S</math> hits the hypotenuse be <math>x</math> and <math>5-x</math>. Also, connect the outermost vertex of <math>S</math> to the vertices that <math>S</math> isn't connected to. Note that the line that hits the hypotenuse must create a right angle, since it is the shortest possible distance. This creates two pairs of right triangles that share the same hypotenuse. This means that we can set up a system of equations using the Pythagorean Theorem:
 +
<cmath>\begin{align*}
 +
a^2+(4-a)^2&=(5-x)^2+2^2, \\
 +
a^2+(3-a)^2&=x^2+2^2.
 +
\end{align*}</cmath>
 +
After *some* algebra, we obtain <math>a=\frac{2}{7}</math>, which gives the answer <math>\boxed{\textbf{(D) } \frac{145}{147}}</math>.
 +
 
 +
==Solution 10 (Proportions)==
 +
We name <i><b>small triangle</b></i> the  triangle similar to given in which unplanted square <math>S</math> is inscribed.
 +
The height of given triangle is 2.4 units so similarity coefficient is <math>\frac {2.4 - 2}{2.4} = \frac {1}{6}</math> , the  area is <math>\frac {1}{36}</math> of total area.
  
−
Since we know <math>m=\frac{4}{3}</math> , we can see that the line rises by <math>\frac{8}{5}</math> and moves to the right by <math>\frac{6}{5}</math> to meet the hypotenuse. (Let <math>2 = 5x</math> and the rise be <math>4x</math> and the run be <math>3x</math> and then solve.) Therefore, line <math>\l</math> intersects the hypotenuse at the point <math>(x+\frac{6}{5}, x+\frac{8}{5})</math>. Plugging into the equation for the hypotenuse we have <math>x=\frac{2}{7}</math> , and after a bit of computation we get <math>\boxed{\textbf{(D) } \frac{145}{147}}</math>
+
The ratio of planted area in <i><b>small triangle</b></i> to the area of the square is <math>\frac {3}{8} + \frac {2}{3} = \frac {25}{24}.</math>
  
 +
The fraction of planted area in <i><b>small triangle</b></i> is <math>\frac {25}{25+24} = \frac {25}{49}.</math>
  
 +
Therefore, the fraction of the planted  field is <math>\frac {25}{49} \cdot  \frac {1}{36} + \frac {35}{36} = \boxed{\textbf{(D) } \frac{145}{147}}.</math>
  
−
==Solution 7(Slightly different from first solution):==
+
'''vladimir.shelomovskii@gmail.com, vvsss'''
−
Same drawing as before:
 
  
 +
==Solution 11 (Bash)==
 
<asy>
 
<asy>
−
draw((0,0)--(4,0)--(0,3)--(0,0));
+
size(240);
−
draw((0,0)--(0.3,0)--(0.3,0.3)--(0,0.3)--(0,0));
+
pair A, B, C, D, F, X, Y, P, Q, M, N;
−
fill(origin--(0.3,0)--(0.3,0.3)--(0,0.3)--cycle, gray);
+
A = origin; label(A, "$A$", SW);
−
label("$4$", (2,0), S);
+
B = (4,0); label(B, "$B$", S);
−
label("$3$", (0,1.5), W);
+
C = (0,3); label(C, "$C$", W);
−
label("$2$", (.8,1), E);
+
D = (2/7,2/7); label(D, "$D$", NE);
−
label("$S$", (0,0), NE);
+
F = foot(D,B,C); label(F, "$F$", NE);
−
draw((0.3,0.3)--(1.4,1.9), dashed);
+
X = (2/7,39/14); label(X, "$X$", NE, red);
−
draw((0.3,0.3)--(4,0), dashed);
+
Y = (76/21,2/7); label(Y, "$Y$", NE, red);
−
draw((0.3,0.3)--(0,3), dashed);
+
P = foot(X,A,C); label(P, "$P$", W, red);
−
label("$\small{a}$", (0.15,0.3), N);
+
Q = foot(Y,A,B); label(Q, "$Q$", S, red);
−
label("$\small{a}$", (0.3,0.15), E);
+
M = (2/7,0); label(M, "$M$", S);
 +
N = (0,2/7); label(N, "$N$", W);
 +
 
 +
fill(A--(2/7,0)--D--(0,2/7)--cycle, lightgray);
 +
draw(A--B--C--cycle);
 +
draw((2/7,0)--D--(0,2/7));
 +
label("$x$", midpoint(A--M), S);
 +
label("$x$", midpoint(A--N), W);
 +
label("$2$", midpoint(D--F), SE);
 +
draw(D--F);
 +
draw(D--X, red);
 +
draw(D--Y, red);
 +
draw(X--P, red);
 +
draw(Y--Q, red);
 
</asy>
 
</asy>
  
−
Let's assign <math>a</math> as the side length of box S. We then get each of the smaller triangle areas.
+
Denote <math>A,B,C</math> to be the three vertices of the triangular field. Also denote <math>A,M,D,N</math> to be the vertices of the square <math>S</math>. Let <math>X</math> be on <math>BC</math> such that <math>AC\parallel DX</math> and <math>Y</math> be on <math>BC</math> such that <math>AB\parallel DY</math>. Let <math>P</math> and <math>Q</math> be the foot of the altitudes from <math>X</math> to <math>AC</math> and from <math>Y</math> to <math>AB</math> respectively.
−
The sum of all the triangular areas(not including the box) is equal to
 
−
<math>\frac{(3-a) \cdot a}{2} + \frac{(4-a) \cdot a}{2} + \frac{5 \cdot 2}{2} = \frac{3 \cdot 4}{2} - a^2</math>
 
−
 
 
−
You can solve for <math>a=\frac{2}{7}</math>
 
−
 
 
−
Then, the ratio would be <math>1-\dfrac{{\frac{2}{7}}^2}{6}</math> which is equal to <math>\boxed{\textbf{(D) } \frac{145}{147}}</math>
 
  
−
~Starshooter11
+
Note that <math>\triangle ABC \sim \triangle DYX \sim \triangle PXC \sim \triangle QBY</math>. Thus, <math>PC = x \cdot \frac34</math> and <math>QB = x \cdot \frac43</math>, making
 +
<cmath>\begin{align*}
 +
DX &= 3-x-\dfrac{3}{4}x = 3-\dfrac{7}{4}x, \\
 +
MQ &= 4-x-\dfrac{4}{3}x = 4-\dfrac{7}{3}x.
 +
\end{align*}</cmath>
 +
Also from the similarity ratio is the fact that <math>CX = \frac54 x</math> and <math>BY = \frac53 x</math>, making
 +
<cmath>XY = 5 - \dfrac{5}{4}x - \dfrac{5}{3}x = 5 - \dfrac{35}{12}x.</cmath>
 +
Computing the area of <math>\triangle XDY</math> in two ways gives an equation for <math>x</math>:
 +
<cmath>\begin{align*}
 +
\left(3-\dfrac{7}{4}x\right)\left(4-\dfrac{7}{3}x\right) &= 2\cdot \left(5 - \dfrac{35}{12}x\right) \\
 +
10-\dfrac{35}{6}x &= \dfrac{49}{12}x^2 - 14x + 12 \\
 +
\dfrac{49}{12}x^2 - \dfrac{49}{6}x + 2 &= 0 \\
 +
49x^2 - 98x + 24 &= 0 \\
 +
x&=\dfrac{2}{7} \text{ or } \dfrac{12}{7}.
 +
\end{align*}</cmath>
 +
But <math>x=\dfrac{12}{7}</math> is extraneous. Thus, the area of square <math>S = x^2 = \dfrac{4}{49}</math>, making the portion of the field that is planted being <cmath>1 - \dfrac{\tfrac{4}{49}}{6} = 1 - \dfrac{2}{147} = \boxed{\textbf{(D) } \frac{145}{147}}.</cmath>
  
−
==Solution 8(Similar but cleaner than first solution):==
+
-Solution by sml1809
  
−
Instead of dividing the large triangle into three right triangles plus a square, simply draw one diagonal of the square (can someone asymptote this for me I don't know how) divide the large triangle into three triangles with the sides <math>3,4,</math> and <math>5</math> as their bases. The total area is <math>6</math> as found above, and the square's side length is <math>x</math>. The area equation is:
+
==Solution 12 ==
−
<cmath>\dfrac{3x}{2}+\dfrac{4x}{2}+\dfrac{5\cdot2}{2}=6,</cmath> which solves to <math>x=\dfrac{2}{7},</math> the same as the first solution (but easier to calculate). The final answer is <math>\boxed{\dfrac{145}{147}}</math>.
+
Let the square have side length <math>x</math>. Note that when <math>x = \dfrac{12}{7}</math>, the square is inscribed and touching the hypotenuse. Denote <math>f(x)</math> to be the minimum distance from the square to the hypotenuse. Notably, <math>f</math> is linear with respect to <math>x</math>. <math>f(0) = \dfrac{12}{5}</math>, because it is simply the length of the hypotenuse's altitude. Similarly, <math>f(\dfrac{12}{7}) = 0</math>. We can find that <math>f(x) = \dfrac{12-7x}{5}</math>. Setting this equal to <math>2</math>, we get that <math>f(\dfrac{2}{7}) = 2</math>. Therefore, the side has side length <math>\dfrac27</math>, and has area <math>\dfrac{4}{49}</math>. So, the unshaded area is <math>1 - \dfrac{\frac{4}{49}}{6} = \boxed{\dfrac{145}{147}}</math>, or <math>(D)</math>. ~Puck_0
  
−
==Hardness of Problem==
+
== Video Solution by Pi Academy (Fast and Easy with Area Addition) ==
 +
https://youtu.be/Jdx74PGIgpw?si=kO6EGrKQQUkp22X1
  
−
This problem is medium hard or on a scale of 10, a 6.5. The problem itself requires the drawing of a few obvious lines and algebra, although the image deceives the solver.
+
~ Pi Academy
  
−
Comment: Hardness and difficulty are relative to the problem solver. It's can be useful in personal situations to label questions with a hardness scale, but it may not suffice for other problem solvers. Please don't feel pressured to spontaneously know concepts that are difficult for you. Additionally, nothing is 'obvious' because it's all relative to the problem solver and individual minds. Happy mathing :P
+
== Video Solution (#21-#25) ==
 +
https://youtube.com/playlist?list=PLpxy89D2tvVow8EoCSsNY3Y-2SwJly_SZ&si=aEJ3Ttjck10aCIUH
  
 
== Video Solution by Richard Rusczyk ==
 
== Video Solution by Richard Rusczyk ==
Line 138: Line 273:
  
 
~ dolphin7
 
~ dolphin7
−
 
  
 
==See Also==
 
==See Also==

Latest revision as of 23:41, 30 October 2024

The following problem is from both the 2018 AMC 10A #23 and 2018 AMC 12A #17, so both problems redirect to this page.

Problem

Farmer Pythagoras has a field in the shape of a right triangle. The right triangle's legs have lengths $3$ and $4$ units. In the corner where those sides meet at a right angle, he leaves a small unplanted square $S$ so that from the air it looks like the right angle symbol. The rest of the field is planted. The shortest distance from $S$ to the hypotenuse is $2$ units. What fraction of the field is planted?

[asy] /* Edited by MRENTHUSIASM */ size(160); pair A, B, C, D, F; A = origin; B = (4,0); C = (0,3); D = (2/7,2/7); F = foot(D,B,C); fill(A--(2/7,0)--D--(0,2/7)--cycle, lightgray); draw(A--B--C--cycle); draw((2/7,0)--D--(0,2/7)); label("$4$", midpoint(A--B), N); label("$3$", midpoint(A--C), E); label("$2$", midpoint(D--F), SE); label("$S$", midpoint(A--D)); draw(D--F, dashed); [/asy]

$\textbf{(A) } \frac{25}{27} \qquad \textbf{(B) } \frac{26}{27} \qquad \textbf{(C) } \frac{73}{75} \qquad \textbf{(D) } \frac{145}{147} \qquad \textbf{(E) } \frac{74}{75}$

Solution 1 (Area Addition)

Note that the hypotenuse of the field is $5,$ and the area of the field is $6.$ Let $x$ be the side-length of square $S.$

We partition the field into a red triangle, a yellow triangle, and a green triangle, as shown below: [asy] /* Edited by MRENTHUSIASM */ size(180); pair A, B, C, D, F; A = origin; B = (4,0); C = (0,3); D = (2/7,2/7); F = foot(D,B,C); fill(A--D--C--cycle, red); fill(A--D--B--cycle, yellow); fill(B--D--C--cycle, green); draw(A--B--C--cycle); label("$5$", midpoint(B--C), NE); label("$4$", midpoint(A--B), S); label("$3$", midpoint(A--C), W); label("$2$", midpoint(D--F), SE); label("$S$", midpoint(A--D)); label("$x$", midpoint((0,2/7)--D), N); label("$x$", midpoint((2/7,0)--D), E); draw((2/7,0)--D--(0,2/7)); draw(A--D^^B--D^^C--D, dashed); draw(D--F, dashed); [/asy] Let the brackets denote areas. By area addition, we set up an equation for $x:$ \begin{align*} [\text{Red Triangle}]+[\text{Yellow Triangle}]+[\text{Green Triangle}]&=[\text{Field}] \\ \frac{3x}{2}+\frac{4x}{2}+\frac{5\cdot2}{2}&=6, \end{align*} from which $x=\frac27.$ Therefore, the answer is \[\frac{[\text{Field}]-[S]}{[\text{Field}]}=\frac{6-x^2}{6}=\boxed{\textbf{(D) } \frac{145}{147}}.\] ~MRENTHUSIASM

Solution 2 (Area Addition)

Let the square have side length $x$. Connect the upper-right vertex of square $S$ with the two vertices of the triangle's hypotenuse. This divides the triangle in several regions whose areas must add up to the area of the whole triangle, which is $6$. [asy] /* Edited by MRENTHUSIASM */ size(180); pair A, B, C, D, F; A = origin; B = (4,0); C = (0,3); D = (2/7,2/7); F = foot(D,B,C); fill(A--(2/7,0)--D--(0,2/7)--cycle, lightgray); draw(A--B--C--cycle); label("$5$", midpoint(B--C), NE); label("$4$", midpoint(A--B), S); label("$3$", midpoint(A--C), W); label("$2$", midpoint(D--F), SE); label("$S$", midpoint(A--D)); label("$x$", midpoint((0,2/7)--D), N); label("$x$", midpoint((2/7,0)--D), E); draw((2/7,0)--D--(0,2/7)); draw(B--D^^C--D, dashed); draw(D--F, dashed); [/asy] Square $S$ has area $x^2$, and the two thin triangle regions have area $\dfrac{x(3-x)}{2}$ and $\dfrac{x(4-x)}{2}$. The final triangular region with the hypotenuse as its base and height $2$ has area $5$. Thus, we have \[x^2+\dfrac{x(3-x)}{2}+\dfrac{x(4-x)}{2}+5=6.\] Solving gives $x=\dfrac{2}{7}$. The area of $S$ is $\dfrac{4}{49}$ and the desired ratio is $\dfrac{6-\tfrac{4}{49}}{6}=\boxed{\textbf{(D) } \frac{145}{147}}$.

Alternatively, once you get $x=\frac{2}{7}$, you can avoid computation by noticing that there is a denominator of $7$, so the answer must have a factor of $7$ in the denominator, which only $\frac{145}{147}$ does.

Solution 3 (Similar Triangles)

Let the square have side length $s$. If we were to extend the sides of the square further into the triangle until they intersect on point on the hypotenuse, we'd have a similar right triangle formed between the hypotenuse and the two new lines, and two smaller similar triangles that share a side of length $2$. Using the side-to-side ratios of these triangles, we can find that the length of the larger similar triangle is $\frac{5}{3}(2)=\frac{10}{3}$. Now, let's extend this larger similar right triangle to the left until it hits the side of length $3$. Now, the length is $\frac{10}{3}+s$, and using the ratios of the side lengths, the height is $\frac{3}{4}\left(\frac{10}{3}+s\right)=\frac{5}{2}+\frac{3s}{4}$. Looking at the diagram, if we add the height of this triangle to the side length of the square, we'd get $3$, so \begin{align*} \frac{5}{2}+\frac{3s}{4}+s&=3 \\ \frac{5}{2}+\frac{7s}{4}&=3 \\ \frac{7s}{4}&=\frac{1}{2} \\ s&=\frac{2}{7}. \end{align*} So, the area of the square is $\left(\frac{2}{7}\right)^2=\frac{4}{49}$.

Now comes the easy part--finding the ratio of the areas: $\frac{3\cdot 4 \cdot \frac{1}{2} -\frac{4}{49}}{3\cdot 4 \cdot \frac{1}{2}}=\frac{6-\frac{4}{49}}{6}=\frac{294-4}{294}=\frac{290}{294}=\boxed{\textbf{(D) } \frac{145}{147}}$.

Solution 4 (Similar Triangles)

[asy] /* Edited by MRENTHUSIASM */ size(180); pair A, B, C, D, F; A = origin; B = (4,0); C = (0,3); D = (2/7,2/7); F = foot(D,B,C); fill(A--(2/7,0)--D--(0,2/7)--cycle, lightgray); draw(A--B--C--cycle); label("$4$", midpoint(A--B), S); label("$3$", midpoint(A--C), W); label("$2$", midpoint(D--F), SE); label("$S$", midpoint(A--D)); label("$\ell$", midpoint((0,2/7)--D), N); label("$\ell$", midpoint((2/7,0)--D), E); label("$\ell$", midpoint((2/7,2/7+5/2)--(0,2/7+5/2)), S); label("$\ell$", midpoint((2/7+10/3,2/7)--(2/7+10/3,0)), W); label("$\frac{5}{2}$", midpoint((2/7,2/7+5/2)--D), E); label("$\frac{10}{3}$", midpoint((2/7+10/3,2/7)--D), N); draw((2/7,0)--D--(0,2/7)); draw((2/7,2/7+5/2)--D^^(2/7+10/3,2/7)--D, dashed); draw((2/7,2/7+5/2)--(0,2/7+5/2)^^(2/7+10/3,2/7)--(2/7+10/3,0), dashed); draw(D--F, dashed); [/asy] On the diagram above, find two smaller triangles similar to the large one with side lengths $3$, $4$, and $5$; consequently, the segments with length $\frac{5}{2}$ and $\frac{10}{3}$.

With $\ell$ being the side length of the square, we need to find an expression for $\ell$. Using the hypotenuse, we can see that $\frac{3}{2}+\frac{8}{3}+\frac{5}{4}\ell+\frac{5}{3}\ell=5$. Simplifying, $\frac{35}{12}\ell=\frac{5}{6}$, or $\ell=\frac27$.

A different calculation would yield $\ell+\frac{3}{4}\ell+\frac{5}{2}=3$, so $\frac{7}{4}\ell=\frac{1}{2}$. In other words, $\ell=\frac{2}{7}$, while to check, $\ell+\frac{4}{3}\ell+\frac{10}{3}=4$. As such, $\frac{7}{3}\ell=\frac{2}{3}$, and $\ell=\frac{2}{7}$.

Finally, we get $A(\Square S)=\ell^2=\frac{4}{49}$, to finish. As a proportion of the triangle with area $6$, the answer would be $1-\frac{4}{49\cdot6}=1-\frac{2}{147}=\frac{145}{147}$, so $\boxed{\textbf{(D) } \frac{145}{147}}$ is correct.

Solution 5 (Similar Triangles)

Let the side length of the square be $x$. First off, let us make a similar triangle with the segment of length $2$ and the top-right corner of $S$. Therefore, the longest side of the smaller triangle must be $2 \cdot \frac54 = \frac52$. We then do operations with that side in terms of $x$. We subtract $x$ from the bottom, and $\frac{3x}{4}$ from the top. That gives us the equation of $3-\frac{7x}{4} = \frac{5}{2}$. Solving, \[12-7x = 10 \implies x = \frac{2}{7}.\] Thus, $x^2 = \frac{4}{49}$, so the fraction of the triangle (area $6$) covered by the square is $\frac{2}{147}$. The answer is then $\boxed{\textbf{(D) } \frac{145}{147}}$.

Solution 6 (Similar Triangles)

[asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */ import graph; size(180); real labelscalefactor = 1.5; /* changes label-to-point distance */ // pen dps = linewidth(0.5) + fontsize(10); // defaultpen(dps); /* default pen style */ // pen dotstyle = black; /* point style */ real xmin = -1.6030465381283199, xmax = 7.095084767820557, ymin = -1.3624649422453508, ymax = 4.065350676871526; /* image dimensions */ /* draw figures */ draw((0,0)--(0,3)); draw((0,0)--(4,0)); draw((4,0)--(0,3)); draw((0,0.2857142857142857)--(0.2857142857142857,0.2857142857142857)); draw((0.2857142857142857,0.2857142857142857)--(0.2857142857142857,0)); draw((0.07142857142857142,0)--(1.4857142857142858,1.885714285714286)); label("$A$",(0, 0),SW*labelscalefactor); label("$B$",(4,0),SE*labelscalefactor); label("$C$",(0, 3),N*labelscalefactor); label("$D$",(0.2857142857142857,0),S*labelscalefactor); label("$E$",(0.2857142857142857, 0.2857142857142857),E*labelscalefactor); label("$F$",(0.0714285714, 0),S*labelscalefactor); label("$G$", (1.49, 1.89), NE*labelscalefactor); /* dots and labels */ clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy] Let $AD=x$. Note that $\triangle DEF$ is a $3{-}4{-}5$ triangle, so $EF=\frac{5}{4}x$ and $FD=\frac{3}{4}x$. $BF=BD+FD=4-x+\frac{3}{4}x=4-\frac{1}{4}x$. We know that $GE$ is $2$ from the problem so $GF=2+\frac{5}{4}x$. $\triangle FGB$ is also a $3{-}4{-}5$ triangle with $GF:BF=3:5$. We now have $3\left(4-\frac{1}{4}x\right)=5\left(2+\frac{5}{4}x\right)$. Solving this equation, we get that $x=\frac{2}{7}$ so the area of $S$ is $\frac{4}{49}$. The area of the triangle is $\frac{3\cdot 4}{2}=6$ so the fraction of field that is unplanted is $\frac{\frac{4}{49}}{6}=\frac{2}{147}$. Thus, the fraction of the field that is planted is $1-\frac{2}{147}=\boxed{\textbf{(D) } \frac{145}{147}}$.

~Heavytoothpaste

Solution 7 (Coordinate Geometry)

We use coordinate geometry. Let the right angle be at $(0,0)$ and the hypotenuse be the line $3x+4y = 12$ for $0\le x\le 3$. Denote the position of $S$ as $(s,s)$, and by the point to line distance formula, we know that \begin{align*} \frac{|3s+4s-12|}{5} &= 2 \\ |7s-12| &= 10 \end{align*} Solving this, we get $s=\frac{22}{7}, \frac{2}{7}$. Obviously $s<\frac{22}{7}$, so $s = \frac{2}{7}$, and from here, the rest of the solution follows to get $\boxed{\textbf{(D) } \frac{145}{147}}$.

Solution 8 (Coordinate Geometry)

Let the right angle be at $(0,0)$, the point $(x,x)$ be the far edge of the unplanted square and the hypotenuse be the line $y=-\frac{3}{4}x+3$. Since the line from $(x,x)$ to the hypotenuse is the shortest possible distance, we know this line, call it line $\l$, is perpendicular to the hypotenuse and therefore has a slope of $\frac{4}{3}$.

Since we know $m=\frac{4}{3}$ , we can see that the line rises by $\frac{8}{5}$ and moves to the right by $\frac{6}{5}$ to meet the hypotenuse. (Let $2 = 5x$ and the rise be $4x$ and the run be $3x$ and then solve.) Therefore, line $\l$ intersects the hypotenuse at the point $\left(x+\frac{6}{5}, x+\frac{8}{5}\right)$. Plugging into the equation for the hypotenuse we have $x=\frac{2}{7}$ , and after a bit of computation we get $\boxed{\textbf{(D) } \frac{145}{147}}$.

Solution 9 (Pythagorean Theorem)

Let the side length of the square be $a$, and the lengths that the line from $S$ hits the hypotenuse be $x$ and $5-x$. Also, connect the outermost vertex of $S$ to the vertices that $S$ isn't connected to. Note that the line that hits the hypotenuse must create a right angle, since it is the shortest possible distance. This creates two pairs of right triangles that share the same hypotenuse. This means that we can set up a system of equations using the Pythagorean Theorem: \begin{align*} a^2+(4-a)^2&=(5-x)^2+2^2, \\ a^2+(3-a)^2&=x^2+2^2. \end{align*} After *some* algebra, we obtain $a=\frac{2}{7}$, which gives the answer $\boxed{\textbf{(D) } \frac{145}{147}}$.

Solution 10 (Proportions)

We name small triangle the triangle similar to given in which unplanted square $S$ is inscribed. The height of given triangle is 2.4 units so similarity coefficient is $\frac {2.4 - 2}{2.4} = \frac {1}{6}$ , the area is $\frac {1}{36}$ of total area.

The ratio of planted area in small triangle to the area of the square is $\frac {3}{8} + \frac {2}{3} = \frac {25}{24}.$

The fraction of planted area in small triangle is $\frac {25}{25+24} = \frac {25}{49}.$

Therefore, the fraction of the planted field is $\frac {25}{49} \cdot \frac {1}{36} + \frac {35}{36} = \boxed{\textbf{(D) } \frac{145}{147}}.$

vladimir.shelomovskii@gmail.com, vvsss

Solution 11 (Bash)

[asy] size(240); pair A, B, C, D, F, X, Y, P, Q, M, N; A = origin; label(A, "$A$", SW); B = (4,0); label(B, "$B$", S); C = (0,3); label(C, "$C$", W); D = (2/7,2/7); label(D, "$D$", NE); F = foot(D,B,C); label(F, "$F$", NE); X = (2/7,39/14); label(X, "$X$", NE, red); Y = (76/21,2/7); label(Y, "$Y$", NE, red); P = foot(X,A,C); label(P, "$P$", W, red); Q = foot(Y,A,B); label(Q, "$Q$", S, red); M = (2/7,0); label(M, "$M$", S); N = (0,2/7); label(N, "$N$", W); fill(A--(2/7,0)--D--(0,2/7)--cycle, lightgray); draw(A--B--C--cycle); draw((2/7,0)--D--(0,2/7)); label("$x$", midpoint(A--M), S); label("$x$", midpoint(A--N), W); label("$2$", midpoint(D--F), SE); draw(D--F); draw(D--X, red); draw(D--Y, red); draw(X--P, red); draw(Y--Q, red); [/asy]

Denote $A,B,C$ to be the three vertices of the triangular field. Also denote $A,M,D,N$ to be the vertices of the square $S$. Let $X$ be on $BC$ such that $AC\parallel DX$ and $Y$ be on $BC$ such that $AB\parallel DY$. Let $P$ and $Q$ be the foot of the altitudes from $X$ to $AC$ and from $Y$ to $AB$ respectively.

Note that $\triangle ABC \sim \triangle DYX \sim \triangle PXC \sim \triangle QBY$. Thus, $PC = x \cdot \frac34$ and $QB = x \cdot \frac43$, making \begin{align*} DX &= 3-x-\dfrac{3}{4}x = 3-\dfrac{7}{4}x, \\ MQ &= 4-x-\dfrac{4}{3}x = 4-\dfrac{7}{3}x. \end{align*} Also from the similarity ratio is the fact that $CX = \frac54 x$ and $BY = \frac53 x$, making \[XY = 5 - \dfrac{5}{4}x - \dfrac{5}{3}x = 5 - \dfrac{35}{12}x.\] Computing the area of $\triangle XDY$ in two ways gives an equation for $x$: \begin{align*} \left(3-\dfrac{7}{4}x\right)\left(4-\dfrac{7}{3}x\right) &= 2\cdot \left(5 - \dfrac{35}{12}x\right) \\ 10-\dfrac{35}{6}x &= \dfrac{49}{12}x^2 - 14x + 12 \\ \dfrac{49}{12}x^2 - \dfrac{49}{6}x + 2 &= 0 \\ 49x^2 - 98x + 24 &= 0 \\ x&=\dfrac{2}{7} \text{ or } \dfrac{12}{7}. \end{align*} But $x=\dfrac{12}{7}$ is extraneous. Thus, the area of square $S = x^2 = \dfrac{4}{49}$, making the portion of the field that is planted being \[1 - \dfrac{\tfrac{4}{49}}{6} = 1 - \dfrac{2}{147} = \boxed{\textbf{(D) } \frac{145}{147}}.\]

-Solution by sml1809

Solution 12

Let the square have side length $x$. Note that when $x = \dfrac{12}{7}$, the square is inscribed and touching the hypotenuse. Denote $f(x)$ to be the minimum distance from the square to the hypotenuse. Notably, $f$ is linear with respect to $x$. $f(0) = \dfrac{12}{5}$, because it is simply the length of the hypotenuse's altitude. Similarly, $f(\dfrac{12}{7}) = 0$. We can find that $f(x) = \dfrac{12-7x}{5}$. Setting this equal to $2$, we get that $f(\dfrac{2}{7}) = 2$. Therefore, the side has side length $\dfrac27$, and has area $\dfrac{4}{49}$. So, the unshaded area is $1 - \dfrac{\frac{4}{49}}{6} = \boxed{\dfrac{145}{147}}$, or $(D)$. ~Puck_0

Video Solution by Pi Academy (Fast and Easy with Area Addition)

https://youtu.be/Jdx74PGIgpw?si=kO6EGrKQQUkp22X1

~ Pi Academy

Video Solution (#21-#25)

https://youtube.com/playlist?list=PLpxy89D2tvVow8EoCSsNY3Y-2SwJly_SZ&si=aEJ3Ttjck10aCIUH

Video Solution by Richard Rusczyk

https://www.youtube.com/watch?v=p9npzq4FY_Y

~ dolphin7

See Also

2018 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 22 		Followed by
Problem 24
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions
2018 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 16 		Followed by
Problem 18
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

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