1.31. The diagonal of an isosceles trapezoid bisects its obtuse angle. The smaller base of the trapezoid is $3 \mathrm{~cm}$, and the perimeter is $42 \mathrm{~cm}$. Find the area of the trapezoid.
1.31. By the condition, $\angle B C A=$ $=\angle A C D$ (Fig. 1.27). But $\angle B C A=$ $=\angle C A D$, so $\triangle A C D$ is isosceles and $A D=C D$. We have $3 A D+B C=42$; since $B C=3$ cm, then $A D=13$ cm. Draw $B K \perp A D$; then $A K=\frac{1}{2}(13-3)=5$ (cm) and from $\triangle A K B$ we find $B K=\sqrt{13^{2}-5^{2}}=$ $=12$ (cm). Therefore,

$$
S=\frac{1}{2}(3+13) \cdot 12=96\left(\mathrm{~cm}^{2}\right)
$$