Difference between revisions of "2021 Fall AMC 10B Problems/Problem 13"

Revision as of 19:31, 29 August 2022

Problem

A square with side length $3$ is inscribed in an isosceles triangle with one side of the square along the base of the triangle. A square with side length $2$ has two vertices on the other square and the other two on sides of the triangle, as shown. What is the area of the triangle?

$[asy] //diagram by kante314 draw((0,0)--(8,0)--(4,8)--cycle, linewidth(1.5)); draw((2,0)--(2,4)--(6,4)--(6,0)--cycle, linewidth(1.5)); draw((3,4)--(3,6)--(5,6)--(5,4)--cycle, linewidth(1.5)); [/asy]$

$(\textbf{A})\: 19\frac14\qquad(\textbf{B}) \: 20\frac14\qquad(\textbf{C}) \: 21 \frac34\qquad(\textbf{D}) \: 22\frac12\qquad(\textbf{E}) \: 23\frac34$

Solution 1

Let's split the triangle down the middle and label it:

$[asy] import olympiad; pair A,B,C,D,E,F,G,H,I,J,K; A = origin; B = (0.5,0); C=(2.5,0); D=(3,0); E = (0.5,2); F=(0.83333333333,2); G=(2.166666666667,2); H=(2.5,2); I=(0.83333333333,3.333333333333); J=(2.166666666667,3.3333333333); K=(1.5,6); draw(A--D--K--cycle); draw(B--E); draw(C--H); draw(F--I); draw(G--J); draw(I--J); draw(E--H); draw(K--(1.5,0)); label("$A$",(1.5,0),S); label("$B$",(1.5,2),SW); label("$C$",(1.5,3.3333333),SW); label("$D$",D,SE); label("$E$",H,SE); label("$F$",J,SE); label("$G$",K,N); [/asy]$

We see that $\bigtriangleup ADG \sim \bigtriangleup BEG \sim \bigtriangleup CFG$ by AA similarity. $BE = \frac{3}{2}$ because $AK$ cuts the side length of the square in half; similarly, $CF = 1$ . Let $CG = h$ : then by side ratios,

$\frac{h+2}{h} = \frac{\frac{3}{2}}{1} \implies 2(h+2) = 3h \implies h = 4$ .

Now the height of the triangle is $AG = 4+2+3 = 9$ . By side ratios, $\frac{9}{4} = \frac{AD}{1} \implies AD = \frac{9}{4}$ .

The area of the triangle is $AG\cdot AD = 9 \cdot \frac{9}{4} = \frac{81}{4} = \boxed{B}$

~KingRavi

Solution 2

By similarity, the height is $3+\frac31\cdot2=9$ and the base is $\frac92\cdot1=4.5$ . Thus the area is $\frac{9\cdot4.5}2=20.25=20\frac14$ , or $\boxed{(\textbf{B})}$ .

~Hefei417, or 陆畅 Sunny from China

Solution 3 (With two different endings)

This solution is based on this figure: Image:2021_AMC_10B_(Nov)_Problem_13,_sol.png

Denote by $O$ the midpoint of $AB$ .

Because $FG = 3$ , $JK = 2$ , $FJ = KG$ , we have $FJ = \frac{1}{2}$ .

We observe $\triangle ADF \sim \triangle FJH$ . Hence, $\frac{AD}{FJ} = \frac{FD}{HJ}$ . Hence, $AD = \frac{3}{4}$ . By symmetry, $BE = AD = \frac{3}{4}$ .

Therefore, $AB = AD + DE + BE = \frac{9}{2}$ .

Because $O$ is the midpoint of $AB$ , $AO = \frac{9}{4}$ .

We observe $\triangle AOC \sim \triangle ADF$ . Hence, $\frac{OC}{DF} = \frac{AO}{AD}$ . Hence, $OC = 9$ .

Therefore, ${\rm Area} \ \triangle ABC = \frac{1}{2} AB \cdot OC = \frac{81}{4} = 20 \frac{1}{4}$ .

Therefore, the answer is $\boxed{\textbf{(B) }20 \frac{1}{4}}$ .

~Steven Chen (www.professorchenedu.com)

Alternatively, we can find the height in a slightly different way.

Following from our finding that the base of the large triangle $AB = \frac{9}{2}$ , we can label the length of the altitude of $\triangle{CHI}$ as $x$ . Notice that $\triangle{CHI} \sim \triangle{CAB}$ . Hence, $\frac{HI}{AB} = \frac{x}{CO}$ . Substituting and simplifying, $\frac{HI}{AB} = \frac{x}{CO} \Rightarrow \frac{2}{\frac{9}{2}} = \frac{x}{x+5} \Rightarrow \frac{x}{x+5} = \frac{4}{9} \Rightarrow x = 4 \Rightarrow CO = 4 + 5 = 9$ . Therefore, the area of the triangle is $\frac{\frac{9}{2} \cdot 9}{2} = \frac{81}{4} = \boxed{\textbf{(B) }20 \frac{1}{4}}$ .

~mahaler

Solution 4 (Coordinates)

For convenience, we will use the image provided in the third solution.

We can set $O$ as the origin.

We know that $FG = 3$ and $JK = 2$ .

We subtract $JK$ from $FG$ and divide by $2$ to get $KG = FJ = \frac{1}{2}$ .

Since $HIKJ$ is a square, we know that $IK = 2$ .

Using rise over run, we find that the slope of $CB$ is $\frac{-2}{0.5} = -4$ .

The coordinates of $I$ are $(1, 5)$ . We plug this in to get the equation of the line that $CB$ runs along: $y = -4x + 9$

We know that the $x-value$ of $C$ is $0$ . Using this, we find that the $y-value$ is $9$ . So the coordinates of $C$ are $(0, 9)$ .

This gives us the height of $\triangle ACB$ : $CO = 9$ .

Now we need to find the coordinates of $B$ .

We know that the $y-value$ is $0$ . Plugging this in, we find $0 = -4x +9$ , or $\frac{9}{4} = x$ .

The coordinates of $B$ are $(\frac{9}{4}, 0)$ .

Since $\triangle ACB$ is symmetrical along $CO$ , we can multiply $CO$ by $OB$ to get $9 \cdot \frac{9}{4} = \frac{81}{4}$

Simplifying, we get $\boxed{\textbf{(B) }20 \frac{1}{4}}$ for the area.

~Achelois

Video Solution by Interstigation

https://www.youtube.com/watch?v=mq4e-s9ENas

Video Solution

https://youtu.be/gxZE3cscswo

~Education, the Study of Everything

@@ Line 91: / Line 91: @@
 ~mahaler
+==Solution 4 (Coordinates)==
+For convenience, we will use the image provided in the third solution.
+We can set <math>O</math> as the origin.
+We know that <math>FG = 3</math> and <math>JK = 2</math>.
+We subtract <math>JK</math> from <math>FG</math> and divide by <math>2</math> to get <math>KG = FJ = \frac{1}{2}</math>.
+Since <math>HIKJ</math> is a square, we know that <math>IK = 2</math>.
+Using rise over run, we find that the slope of <math>CB</math> is <math>\frac{-2}{0.5} = -4</math>.
+The coordinates of <math>I</math> are <math>(1, 5)</math>. We plug this in to get the equation of the line that <math>CB</math> runs along: <cmath>y = -4x + 9</cmath>
+We know that the <math>x-value</math> of <math>C</math> is <math>0</math>. Using this, we find that the <math>y-value</math> is <math>9</math>. So the coordinates of <math>C</math> are <math>(0, 9)</math>.
+This gives us the height of <math>\triangle ACB</math>: <math>CO = 9</math>.
+Now we need to find the coordinates of <math>B</math>.
+We know that the <math>y-value</math> is <math>0</math>. Plugging this in, we find <math>0 = -4x +9</math>, or <math>\frac{9}{4} = x</math>.
+The coordinates of <math>B</math> are <math>(\frac{9}{4}, 0)</math>.
+Since <math>\triangle ACB</math> is symmetrical along <math>CO</math>, we can multiply <math>CO</math> by <math>OB</math> to get <cmath>9 \cdot \frac{9}{4} = \frac{81}{4}</cmath>
+Simplifying, we get <math>\boxed{\textbf{(B) }20 \frac{1}{4}}</math> for the area.
+~Achelois
 ==Video Solution by Interstigation==

2021 Fall AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 12	Followed by Problem 14
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