# 1988 AJHSME Problems/Problem 9

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## Problem

An isosceles triangle is a triangle with two sides of equal length. How many of the five triangles on the square grid below are isosceles? $[asy] for(int a=0; a<12; ++a) { draw((a,0)--(a,6)); } for(int b=0; b<7; ++b) { draw((0,b)--(11,b)); } draw((0,6)--(2,6)--(1,4)--cycle,linewidth(3)); draw((3,4)--(3,6)--(5,4)--cycle,linewidth(3)); draw((0,1)--(3,2)--(6,1)--cycle,linewidth(3)); draw((7,4)--(6,6)--(9,4)--cycle,linewidth(3)); draw((8,1)--(9,3)--(10,0)--cycle,linewidth(3)); [/asy]$ $\text{(A)}\ 1 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 3 \qquad \text{(D)}\ 4 \qquad \text{(E)}\ 5$

## Solution

The first triangle has two legs of length $\sqrt{5}$, the second has two legs of length 2, the leg lengths of the third triangle are $2$, $\sqrt{5}$, and $\sqrt{13}$, two legs of the fourth triangle have length $\sqrt{10}$, and two legs of the fifth triangle have length $\sqrt{5}$. Therefore all of the triangles in the diagram except the third are isosceles, and there are $4\Rightarrow \mathrm{(D)}$ are isosceles.

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