# 2018 AMC 10A Problems/Problem 23

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

~ dolphin7

## See Also

 2018 AMC 10A (Problems • Answer Key • Resources) Preceded byProblem 22 Followed byProblem 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 byProblem 16 Followed byProblem 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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