2. (15 points) The bisectors of the obtuse angles at the base of a trapezoid intersect on the other base of the trapezoid. Find the area of the trapezoid if its height is 12 cm, and the lengths of the bisectors are 15 cm and $13 \mathrm{~cm}$.
Solution. Let segment $B K$ be the height of the given trapezoid $A B C D$ ( $\mathrm{BK}=\mathrm{CH}=12$ ), $\mathrm{BM}$ and $\mathrm{CM}$ be the bisectors of angles $\angle ABC$ and $\angle BCD$, respectively, with $\mathrm{BM}=15, \mathrm{CM}=13$. In the right triangles $B K M$ and $C H M$, by the Pythagorean theorem, we find respectively:

$$
\begin{aligned}
& K M=\sqrt{B M^{2}-B K^{2}}=\sqrt{15^{2}-12^{2}}=9 \\
& M H=\sqrt{C M^{2}-C H^{2}}=\sqrt{13^{2}-12^{2}}=5
\end{aligned}
$$

![](https://cdn.mathpix.com/cropped/2024_05_06_8c1ab4103bc5f9777fe4g-07.jpg?height=277&width=574&top_left_y=284&top_left_x=1392)

Let $H D=x, A K=y$.

Since $\angle A B M=\angle C B M$ ( $B M$ is the bisector of $\angle A B C$ ) and $\angle C B M=\angle A M B$ (as alternate interior angles when $B C$ and $A D$ are parallel and $B M$ is the transversal), then $\angle A B M=\angle A M B$, meaning $\triangle A B M$ is isosceles. Therefore, $A B=A M=K M+A K=9+y$. In the right triangle $\triangle A B K$, we have: $A B^{2}=A K^{2}+B K^{2}$ or $(9+y)^{2}=y^{2}+144 \Rightarrow 2 y=7 \Rightarrow y=3.5$.

Then $A B=A M=9+3.5=12.5$.

Similarly, since $\angle B C M=\angle M C D$ ( $C M$ is the bisector of $\angle B C D$ ) and $\angle B C M=\angle C M D$ (as alternate interior angles when $B C$ and $A D$ are parallel), then $\angle M C D=\angle C M D$, meaning $\triangle C M D$ is isosceles, with $C D=M D=M H+H D=5+x$.

In the right triangle $\triangle C H D$, we have $C D^{2}=C H^{2}+H D^{2}$ or $(5+x)^{2}=x^{2}+144 \Rightarrow 10 x=119 \Rightarrow x=11.9$. Then: $C D=M D=5+11.9=16.9$.

We get: $A D=A M+M D=12.5+16.9=29.4 ; B C=K H=5+9=14$.

Now we find the area of the trapezoid: $S_{A B C D}=\frac{A D+B C}{2} \cdot C H=\frac{29.4+14}{2} \cdot 12=260.4$ (cm²).

Answer: 260.4.

| Criterion | Points |
| :--- | :---: |
| The correct answer is obtained with justification. | 15 |
| The solution contains a computational error, possibly leading to an incorrect answer, but the correct sequence of all steps is present. | 10 |
| In the solution of the problem, one or two formulas are correctly written, which could be the beginning of a possible solution. | 5 |
| The solution does not meet any of the criteria listed above. | 0 |
| Maximum score | 15 |