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Difference between revisions of "2013 AMC 10A Problems/Problem 16"

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[[Category: Introductory Geometry Problems]]
 
[[Category: Introductory Geometry Problems]]
  
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==Solution==
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==Solution 1==
  
 
Let <math>A</math> be at <math>(6, 5)</math>, B be at <math>(8, -3)</math>, and <math>C</math> be at <math>(9, 1)</math>.  Reflecting over the line <math>x=8</math>, we see that <math>A' = D =  (10,5)</math>, <math>B' = B</math> (as the x-coordinate of B is 8), and <math>C' = E =  (7, 1)</math>.  Line <math>AB</math> can be represented as <math>y=-4x+29</math>, so we see that <math>E</math> is on line <math>AB</math>.   
 
Let <math>A</math> be at <math>(6, 5)</math>, B be at <math>(8, -3)</math>, and <math>C</math> be at <math>(9, 1)</math>.  Reflecting over the line <math>x=8</math>, we see that <math>A' = D =  (10,5)</math>, <math>B' = B</math> (as the x-coordinate of B is 8), and <math>C' = E =  (7, 1)</math>.  Line <math>AB</math> can be represented as <math>y=-4x+29</math>, so we see that <math>E</math> is on line <math>AB</math>.   
  
 
<asy>
 
<asy>
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pair A = (6, 5), B = (8, -3), C = (9, 1), D = (10, 5), E = (7, 1);
+
pair A = (6, 5), B = (8, -3), C = (9, 1), D = (10, 5), E = (7, 1), F = (8, 7/3);
 
draw(A--B--C--cycle^^D--E--B--cycle);
 
draw(A--B--C--cycle^^D--E--B--cycle);
−
dot(A^^B^^C^^D^^E);
+
dot(A^^B^^C^^D^^E^^F);
 
label("$A$",A,NW);
 
label("$A$",A,NW);
 
label("$B$",B,S);
 
label("$B$",B,S);
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label("$D$",D,NE);
 
label("$D$",D,NE);
 
label("$E$",E,W);
 
label("$E$",E,W);
 +
label("$F$", F, N);
  
  
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We see that if we connect <math>A</math> to <math>D</math>, we get a line of length <math>4</math> (between <math>(6, 5)</math> and <math>(10,5)</math>).  The area of <math>\triangle ABD</math> is equal to <math>\frac{bh}{2} = \frac{4(8)}{2} = 16</math>.
 
We see that if we connect <math>A</math> to <math>D</math>, we get a line of length <math>4</math> (between <math>(6, 5)</math> and <math>(10,5)</math>).  The area of <math>\triangle ABD</math> is equal to <math>\frac{bh}{2} = \frac{4(8)}{2} = 16</math>.
  
−
Now, let the point of intersection between <math>AC</math> and <math>DE</math> be <math>F</math>.  If we can just find the area of <math>\triangle ADF</math> and subtract it from 16, we are done.
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Now, let the point of intersection between <math>AC</math> and <math>DE</math> be <math>F</math>.  If we can just find the area of <math>\triangle ADF</math> and subtract it from <math>16</math>, we are done.
  
 
We realize that because the diagram is symmetric over <math>x = 8</math>, the intersection of lines <math>AC</math> and <math>DE</math> should intersect at an x-coordinate of <math>8</math>.  We know that the slope of <math>DE</math> is <math>\frac{5-1}{10-7} = \frac{4}{3}</math>.  Thus, we can represent the line going through <math>E</math> and <math>D</math> as <math>y - 1=\frac{4}{3}(x - 7)</math>.  Plugging in <math>x = 8</math>, we find that the y-coordinate of F is <math>\frac{7}{3}</math>.  Thus, the height of <math>\triangle ADF</math> is <math>5 - \frac{7}{3} = \frac{8}{3}</math>.  Using the formula for the area of a triangle, the area of <math>\triangle ADF</math> is <math>\frac{16}{3}</math>.   
 
We realize that because the diagram is symmetric over <math>x = 8</math>, the intersection of lines <math>AC</math> and <math>DE</math> should intersect at an x-coordinate of <math>8</math>.  We know that the slope of <math>DE</math> is <math>\frac{5-1}{10-7} = \frac{4}{3}</math>.  Thus, we can represent the line going through <math>E</math> and <math>D</math> as <math>y - 1=\frac{4}{3}(x - 7)</math>.  Plugging in <math>x = 8</math>, we find that the y-coordinate of F is <math>\frac{7}{3}</math>.  Thus, the height of <math>\triangle ADF</math> is <math>5 - \frac{7}{3} = \frac{8}{3}</math>.  Using the formula for the area of a triangle, the area of <math>\triangle ADF</math> is <math>\frac{16}{3}</math>.   
  
 
To get our final answer, we must subtract this from <math>16</math>.  <math>[ABD] - [ADF] = 16 - \frac{16}{3} = \boxed{\textbf{(E) }\frac{32}{3}}</math>
 
To get our final answer, we must subtract this from <math>16</math>.  <math>[ABD] - [ADF] = 16 - \frac{16}{3} = \boxed{\textbf{(E) }\frac{32}{3}}</math>
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==Solution 2==
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First, realize that <math>E</math> is the midpoint of <math>AB</math> and <math>C</math> is the midpoint of <math>BD</math>. Connect <math>A</math> to <math>D</math> to form <math>\triangle ABD</math>. Let the midpoint of <math>AD</math> be <math>G</math>. Connect <math>B</math> to <math>G</math>. <math>BG</math> is a median of <math>\triangle ABD</math>.
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 +
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Because <math>\triangle ABD</math> is isosceles, <math>BG</math> is also an altitude of <math>\triangle ABD</math>. We know the length of <math>AD</math> and <math>BG</math> from the given coordinates. The area of <math>\triangle ABD</math> is <math>\frac{bh}{2} = \frac{4(8)}{2} = 16</math>.
 +
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Let the intesection of <math>AC</math>, <math>DE</math> and <math>BG</math> be <math>F</math>. <math>F</math> is the centroid of <math>\triangle ABD</math>. Therefore, it splits <math>BG</math> into <math>BF={2 \over 3}(BG)</math> and <math>FG={1\over 3}(BG)</math>. The area of quadrilateral <math>ABDF = 16\cdot {2 \over 3} = \boxed{\textbf{(E) }\frac{32}{3}}</math>
 +
 +
~Zeric Hang
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==Solution 3==
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Since we have to find the area on a coordinate plane, we can use the [[Shoelace_Theorem|Shoelace Theorem]] to find the area of the intersection. When you reflect it, it makes a quadrilateral.
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<asy>
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pair A = (6, 5), B = (8, -3), C = (9, 1), D = (10, 5), E = (7, 1), F = (8, 7/3);
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draw(A--B--C--cycle^^D--E--B--cycle);
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dot(A^^B^^C^^D^^E^^F);
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label("$A$",A,NW);
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label("$B$",B,S);
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label("$C$",C,SE);
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label("$D$",D,NE);
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label("$E$",E,W);
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label("$F$", F, N);
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 +
 +
</asy>
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 +
Since it is reflected around <math>x=8</math>, the point <math>(8,-3)</math> remains the same on both. The top right corners are <math>(6,5)</math>, and its reflection <math>(10,5)</math>. Now to find the 4th point, point F, we can use the equation of the line DE(<math>\frac{4}{3}x - \frac{25}{3}</math>, and substitute <math>x=8</math>, to get <math>\frac{7}{3}</math>. Now we can use the theorem: <math>\frac{1}{2}(((6*-3)+(10*5)+(8*5)+(\frac{7}{3}*8))-((8*5)+(6*5)+(10*\frac{7}{3}) + (8* -3))) = \boxed{\textbf{(E) }\frac{32}{3}}</math>
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 +
~idk12345678
  
 
==See Also==
 
==See Also==

Latest revision as of 15:47, 3 April 2024

Problem

A triangle with vertices $(6, 5)$, $(8, -3)$, and $(9, 1)$ is reflected about the line $x=8$ to create a second triangle. What is the area of the union of the two triangles?

$\textbf{(A)}\ 9 \qquad\textbf{(B)}\ \frac{28}{3} \qquad\textbf{(C)}\ 10 \qquad\textbf{(D)}\ \frac{31}{3} \qquad\textbf{(E)}\ \frac{32}{3}$

Solution 1

Let $A$ be at $(6, 5)$, B be at $(8, -3)$, and $C$ be at $(9, 1)$. Reflecting over the line $x=8$, we see that $A' = D = (10,5)$, $B' = B$ (as the x-coordinate of B is 8), and $C' = E = (7, 1)$. Line $AB$ can be represented as $y=-4x+29$, so we see that $E$ is on line $AB$.

[asy] pair A = (6, 5), B = (8, -3), C = (9, 1), D = (10, 5), E = (7, 1), F = (8, 7/3); draw(A--B--C--cycle^^D--E--B--cycle); dot(A^^B^^C^^D^^E^^F); label("$A$",A,NW); label("$B$",B,S); label("$C$",C,SE); label("$D$",D,NE); label("$E$",E,W); label("$F$", F, N); [/asy]


We see that if we connect $A$ to $D$, we get a line of length $4$ (between $(6, 5)$ and $(10,5)$). The area of $\triangle ABD$ is equal to $\frac{bh}{2} = \frac{4(8)}{2} = 16$.

Now, let the point of intersection between $AC$ and $DE$ be $F$. If we can just find the area of $\triangle ADF$ and subtract it from $16$, we are done.

We realize that because the diagram is symmetric over $x = 8$, the intersection of lines $AC$ and $DE$ should intersect at an x-coordinate of $8$. We know that the slope of $DE$ is $\frac{5-1}{10-7} = \frac{4}{3}$. Thus, we can represent the line going through $E$ and $D$ as $y - 1=\frac{4}{3}(x - 7)$. Plugging in $x = 8$, we find that the y-coordinate of F is $\frac{7}{3}$. Thus, the height of $\triangle ADF$ is $5 - \frac{7}{3} = \frac{8}{3}$. Using the formula for the area of a triangle, the area of $\triangle ADF$ is $\frac{16}{3}$.

To get our final answer, we must subtract this from $16$. $[ABD] - [ADF] = 16 - \frac{16}{3} = \boxed{\textbf{(E) }\frac{32}{3}}$

Solution 2

First, realize that $E$ is the midpoint of $AB$ and $C$ is the midpoint of $BD$. Connect $A$ to $D$ to form $\triangle ABD$. Let the midpoint of $AD$ be $G$. Connect $B$ to $G$. $BG$ is a median of $\triangle ABD$.


Because $\triangle ABD$ is isosceles, $BG$ is also an altitude of $\triangle ABD$. We know the length of $AD$ and $BG$ from the given coordinates. The area of $\triangle ABD$ is $\frac{bh}{2} = \frac{4(8)}{2} = 16$.


Let the intesection of $AC$, $DE$ and $BG$ be $F$. $F$ is the centroid of $\triangle ABD$. Therefore, it splits $BG$ into $BF={2 \over 3}(BG)$ and $FG={1\over 3}(BG)$. The area of quadrilateral $ABDF = 16\cdot {2 \over 3} = \boxed{\textbf{(E) }\frac{32}{3}}$

~Zeric Hang

Solution 3

Since we have to find the area on a coordinate plane, we can use the Shoelace Theorem to find the area of the intersection. When you reflect it, it makes a quadrilateral.

[asy] pair A = (6, 5), B = (8, -3), C = (9, 1), D = (10, 5), E = (7, 1), F = (8, 7/3); draw(A--B--C--cycle^^D--E--B--cycle); dot(A^^B^^C^^D^^E^^F); label("$A$",A,NW); label("$B$",B,S); label("$C$",C,SE); label("$D$",D,NE); label("$E$",E,W); label("$F$", F, N); [/asy]

Since it is reflected around $x=8$, the point $(8,-3)$ remains the same on both. The top right corners are $(6,5)$, and its reflection $(10,5)$. Now to find the 4th point, point F, we can use the equation of the line DE($\frac{4}{3}x - \frac{25}{3}$, and substitute $x=8$, to get $\frac{7}{3}$. Now we can use the theorem: $\frac{1}{2}(((6*-3)+(10*5)+(8*5)+(\frac{7}{3}*8))-((8*5)+(6*5)+(10*\frac{7}{3}) + (8* -3))) = \boxed{\textbf{(E) }\frac{32}{3}}$

~idk12345678

See Also

2013 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 15 		Followed by
Problem 17
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

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