# Difference between revisions of "2012 AMC 10A Problems/Problem 15"

## Problem

Three unit squares and two line segments connecting two pairs of vertices are shown. What is the area of $\triangle ABC$?

$[asy] unitsize(2cm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4; pair A=(0,0), B=(1,0); pair C=(0.8,-0.4); draw(A--(2,0)); draw((0,-1)--(2,-1)); draw((0,-2)--(1,-2)); draw(A--(0,-2)); draw(B--(1,-2)); draw((2,0)--(2,-1)); draw(A--(2,-1)); draw(B--(0,-2)); pair[] ps={A,B,C}; dot(ps); label("A",A,N); label("B",B,N); label("C",C,W); [/asy]$

$\textbf{(A)}\ \frac16 \qquad\textbf{(B)}\ \frac15 \qquad\textbf{(C)}\ \frac29 \qquad\textbf{(D)}\ \frac13 \qquad\textbf{(E)}\ \frac{\sqrt{2}}{4}$

## Coordinate geo

Making the bottom right corner the origin, the $C$ will be the intersection of $y=2x$ and $y=-\frac{x}{2}+2$ solving for $y$ we get $\frac{8}{5}$. This means that the height of the triangle is $\frac{2}{5}$. Because the base is 1 the area is $\frac{1}{5}$.

## Solution 1

$[asy] unitsize(2cm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4; pair A=(0,0), B=(1,0); pair C=(0.8,-0.4); pair D=(1,-2), E=(0,-2); draw(A--(2,0)); draw((0,-1)--(2,-1)); draw(E--D); draw(A--E); draw(B--D); draw((2,0)--(2,-1)); draw(A--(2,-1)); draw(B--E); pair[] ps={A,B,C,D,E}; dot(ps); label("A",A,N); label("B",B,N); label("C",C,W); label("D",D,S); label("E",E,S); label("1",(D--E),S); label("1",(A--B),N); label("2",(A--E),W); label("\sqrt{5}",(B--E),NW); [/asy]$

$AC$ intersects $BC$ at a right angle, so $\triangle ABC \sim \triangle BED$. The hypotenuse of right triangle BED is $\sqrt{1^2+2^2}=\sqrt{5}$.

$$\frac{AC}{BC}=\frac{BD}{ED} \Rightarrow \frac{AC}{BC} = \frac21 \Rightarrow AC=2BC$$

$$\frac{AC}{AB}=\frac{BD}{BE} \Rightarrow \frac{AC}{1}=\frac{2}{\sqrt{5}} \Rightarrow AC=\frac{2}{\sqrt{5}}$$

Since $AC=2BC$, $BC=\frac{1}{\sqrt{5}}$. $\triangle ABC$ is a right triangle so the area is just $\frac12 \cdot AC \cdot BC = \frac12 \cdot \frac{2}{\sqrt{5}} \cdot \frac{1}{\sqrt{5}} = \boxed{\textbf{(B)}\ \frac15}$

## Solution 2

$[asy] unitsize(2cm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4; pair A=(0,0), B=(1,0); pair C=(0.8,-0.4); pair D=(1,-2), E=(0,-2); pair F=(2,-1); pair G=(0.8,0); draw(A--(2,0)); draw((0,-1)--F); draw(E--D); draw(A--E); draw(B--D); draw((2,0)--F); draw(A--F); draw(B--E); draw(C--G); pair[] ps={A,B,C,D,E,F,G}; dot(ps); label("A",A,N); label("B",B,N); label("C",C,W); label("D",D,S); label("E",E,S); label("F",F,E); label("G",G,N); [/asy]$

Let $\text{E}$ be the origin. Then, $\text{D}=(1, 0)$ $\text{A}=(0, 2)$ $\text{B}=(1, 2)$ $\text{F}=(2, 1)$

${EB}$ can be represented by the line $y=2x$ Also, ${AF}$ can be represented by the line $y=-\frac{1}{2}x+2$

Subtracting the second equation from the first gives us $\frac{5}{2}x-2=0$. Thus, $x=\frac{4}{5}$. Plugging this into the first equation gives us $y=\frac{8}{5}$.

Since $\text{C} (0.8, 1.6)$, $G$ is $(0.8, 2)$,

${AB}=1$ and ${CG}=0.4$.

Thus, $[ABC]=\frac{1}{2} \cdot {AB} \cdot {CG}=\frac{1}{2} \cdot 1 \cdot 0.4=0.2=\frac{1}{5}$. The answer is $\boxed{\textbf{(B)}\ \frac15}$.

## Solution 3

$[asy] unitsize(2cm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4; pair A=(0,0), B=(1,0); pair C=(0.8,-0.4); pair D=(1,-2), E=(0,-2); pair F=(2,-1); pair G=(0.8,0); pair H=(0,-1), I=(0.5,-1); draw(A--(2,0)); draw((0,-1)--F); draw(E--D); draw(A--E); draw(B--D); draw((2,0)--F); draw(A--F); draw(B--E); draw(C--G); draw(H--I); pair[] ps={A,B,C,D,E,F,G, H, I}; dot(ps); label("A",A,N); label("B",B,N); label("C",C,W); label("D",D,S); label("E",E,S); label("F",F,E); label("G",G,N); label("H",H,W); label("I",I,E); [/asy]$

Triangle $EAB$ is similar to triangle $EHI$; line $HI = 1/2$

Triangle $ACB$ is similar to triangle $FCI$ and the ratio of line $AB$ to line $IF = 1 : \frac{3}{2} = 2: 3$.

Based on similarity the length of the height of $GC$ is thus $\frac{2}{5}\cdot1 = \frac{2}{5}$.

Thus, $[ABC]=\frac{1}{2} \cdot {AB} \cdot {CG}=\frac{1}{2} \cdot 1 \cdot \frac{2}{5}=\frac{1}{5}$. The answer is $\boxed{\textbf{(B)}\ \frac15}$

## Solution 4

Let $L$ be the point where the diagonal and the end of the unit square meet, on the right side of the diagram. Let $K$ be the top right corner of the top right unit square, where segment $ABL$ is 2 units in length.

Because of the Pythagorean Theorem, since $AC = 2$ and $LK$ = 1, the diagonal of triangle $ALK$ is $\sqrt{5}$.

Triangle $ALK$ is clearly a similar triangle to triangle $ABC$. Segment $AB$ is the hypotenuse of triangle $ABC$. So, we can write down:

$$AK/AB = LK/BC$$, which is equal to: $$\frac{\sqrt{5}}{1} = \frac{1}{BC}$$ Solving this equation yields:

$$BC = \frac{1}{\sqrt{5}}$$

By Pythagorean theorem, we can now find segment $AC$ $$(\frac{1}{\sqrt{5}})^2 + AC^2 = 1$$ Solving this yields:

$$AC^2 = \frac{4}{5}$$, so $AC = \frac{2}{\sqrt{5}}$

So then we can use $$A = \frac{1}{2} * b * a.$$ So $$A = \frac{1}{2} * \frac{1}{\sqrt{5}} * \frac{2}{\sqrt{5}}$$

$$= \boxed{\textbf{(B)}\ \frac15}$$