1988 AJHSME Problems/Problem 9

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.

1988 AJHSME (Problems • Answer Key • Resources)
Preceded by Problem 8	Followed by Problem 10
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AJHSME/AMC 8 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

Page

Toolbox

Search

1988 AJHSME Problems/Problem 9

Problem

Solution

See Also