Question

The area of triangle $XYZ$ is $8$ square inches. Points $A$ and $B$ are midpoints of congruent segments $\overline{XY}$ and $\overline{XZ}.$ Altitude $\overline{XC}$ bisects $\overline{YZ}.$ The area (in square inches) of the shaded region is

$1 \frac{1}{2}$

$2$

$2 \frac{1}{2}$

$3$

$3 \frac{1}{2}$

Solution:

Since $XY=XZ$ and $XC$ bisects $YZ$ , altitude $XC$ splits $\triangle XYZ$ into two congruent triangles. The left half $\triangle XYC$ has area $4$ .

In $\triangle XYC$ , point $A$ is the midpoint of $XY$ . The horizontal segment through $A$ meets $XC$ halfway up, so the small top triangle is similar to $\triangle XYC$ with scale factor $1/2$ . Its area is therefore $1/4$ of $4$ , or $1$ .

The shaded region is the rest of the left half, so its area is $4-1=3$ .

Thus, D is the correct answer.

2002 AMC 8 Problem 20

Solution:

Problem 20 in Other Years