Difference between revisions of "2020 AMC 10A Problems/Problem 12"

Revision as of 23:21, 31 January 2020

Problem

Triangle $AMC$ is isoceles with $AM = AC$ . Medians $\overline{MV}$ and $\overline{CU}$ are perpendicular to each other, and $MV=CU=12$ . What is the area of $\triangle AMC?$

$\textbf{(A) } 48 \qquad \textbf{(B) } 72 \qquad \textbf{(C) } 96 \qquad \textbf{(D) } 144 \qquad \textbf{(E) } 192$

Solution

Since quadrilateral $UVCM$ has perpendicular diagonals, its area can be found as half of the product of the length of the diagonals. Also note that $\triangle AUV$ has $\frac 14$ the area of triangle $AMC$ by similarity, so $[UVCM]=\frac 34\cdot [AMC].$ Thus, $\frac 12 \cdot 12\cdot 12=\frac 34 \cdot [AMC]$ $72=\frac 34\cdot [AMC]$ $[AMC]=96\rightarrow \boxed{\textbf{(C)}}.$

Solution 2 (CHEATING)

Draw a to-scale diagram with your graph paper and straightedge. Measure the height and approximate the area.

Solution 3 (Trapezoid)

We know that $\triangle AUV \sim \triangle AMC$ , and since the ratios of its sides are $\frac{1}{2}$ , the ratio of of their areas is $(\frac{1}{2})^2=\frac{1}{4}$ .

If $\triangle AUV$ is $\frac{1}{4}$ the area of $\triangle AMC$ , then trapezoid $MUVC$ is $\frac{3}{4}$ the area of $\triangle AMC$ .

Let's call the intersection of $\overline{UC}$ and $\overline{MV}$ $P$ . Let $\overline{UP}=x$ . Then $\overline{PC}=12-x$ . Since $\overline{UC} \perp \overline{MV}$ , $\overline{UP}$ and $\overline{CP}$ are heights of triangles $\triangle MUV$ and $\triangle MCV$ , respectively. Both of these triangles have base $12$ .

Area of $\triangle MUV = \frac{x\cdot12}{2}=6x$

Area of $\triangle MCV = \frac{(12-x)\cdot12}{2}=72-6x$

Adding these two gives us the area of trapezoid $MUVC$ , which is $6x+(72-6x)=72$ .

This is $\frac{3}{4}$ of the triangle, so the area of the triangle is $\frac{4}{3}\cdot{72}=\boxed{\textbf{(C) } 96}$ ~quacker88

Video Solution

https://youtu.be/ZGwAasE32Y4

~IceMatrix

@@ Line 14: / Line 14: @@
 ==Solution 2 (CHEATING)==
 Draw a to-scale diagram with your graph paper and straightedge. Measure the height and approximate the area.
+==Solution 3 (Trapezoid)==
+We know that <math>\triangle AUV \sim \triangle AMC</math>, and since the ratios of its sides are <math>\frac{1}{2}</math>, the ratio of of their areas is <math>(\frac{1}{2})^2=\frac{1}{4}</math>.
+If <math>\triangle AUV</math> is <math>\frac{1}{4}</math> the area of <math>\triangle AMC</math>, then trapezoid <math>MUVC</math> is <math>\frac{3}{4}</math> the area of <math>\triangle AMC</math>.
+Let's call the intersection of <math>\overline{UC}</math> and <math>\overline{MV}</math> <math>P</math>. Let <math>\overline{UP}=x</math>. Then <math>\overline{PC}=12-x</math>. Since <math>\overline{UC}
+\perp \overline{MV}</math>, <math>\overline{UP}</math> and <math>\overline{CP}</math> are heights of triangles <math>\triangle MUV</math> and <math>\triangle MCV</math>, respectively. Both of these triangles have base <math>12</math>.
+Area of <math>\triangle MUV = \frac{x\cdot12}{2}=6x</math>
+Area of <math>\triangle MCV = \frac{(12-x)\cdot12}{2}=72-6x</math>
+Adding these two gives us the area of trapezoid <math>MUVC</math>, which is <math>6x+(72-6x)=72</math>.
+This is <math>\frac{3}{4}</math> of the triangle, so the area of the triangle is <math>\frac{4}{3}\cdot{72}=\boxed{\textbf{(C) } 96}</math> ~quacker88
 ==Video Solution==

2020 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 11	Followed by Problem 13
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