CEMC Fryer, 2011

In any isosceles triangle $ABC$ with $AB = AC$, the altitude $AD$ bisects the base $BC$ so that $BD = DC$.

(a) (i) As shown in $\triangle ABC$, $AB = AC = 25$ and $BC = 14$. Determine the length of the altitude $AD$.

(ii) Determine the area of $\triangle ABC$.

(b) Triangle $ABC$ from part (a) is cut along its altitude from $A$ to $D$ (Figure 1). Each of the two new triangles is then rotated $90^\circ$ about point $D$ until $B$ meets $C$ directly below $D$ (Figure 2). This process creates the new triangle which is labelled $PQR$ (Figure 3).

(i) In $\triangle PQR$, determine the length of the base $PR$.

(ii) Determine the area of $\triangle PQR$.

(c) There are two different isosceles triangles whose side lengths are integers and whose areas are 120. One of these two triangles, $\triangle XYZ$, is shown. Determine the lengths of the three sides of the second triangle.