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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>.   
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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.
+
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>.   

Revision as of 20:46, 28 December 2019

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); draw(A--B--C--cycle^^D--E--B--cycle); dot(A^^B^^C^^D^^E); label("$A$",A,NW); label("$B$",B,S); label("$C$",C,SE); label("$D$",D,NE); label("$E$",E,W); [/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

See Also

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

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