Let $\angle A B D=\angle A C D=\alpha, \angle B A C=\angle B D C=\beta$. Express the square of their common side $B C$ from triangles $A B D$ and $A C D$ and find $\cos \alpha$. Similarly, find $\cos \beta$.

## Solution

Let $\angle A B D=\angle A C D=\alpha, \angle B A C=\angle B D C=\beta$. By the Law of Cosines in triangles $A B D$ and $A C D$,

$$
\begin{gathered}
A D^{2}=9^{2}+8^{2}-2 \cdot 9 \cdot 8 \cdot \cos \alpha=145-144 \cos \alpha \text { and } \\
A D^{2}=7^{2}+4^{2}-2 \cdot 7 \cdot 4 \cdot \cos \alpha=65-56 \cos \alpha .
\end{gathered}
$$

From the equation $145-144 \cos \alpha=65-56 \cos \alpha$, we find that $\cos \alpha=\frac{10}{11}$. Then $\sin \alpha=\sqrt{1-\cos ^{2} \alpha}=\frac{\sqrt{21}}{11}$.

Therefore,

$$
S_{\triangle \mathrm{ABD}}=\frac{1}{2} \cdot A B \cdot B D \cdot \sin \alpha=\frac{1}{2} \cdot 9 \cdot 8 \cdot \frac{\sqrt{21}}{11}=\frac{36 \sqrt{21}}{11}
$$

By the Law of Cosines in triangles $B A C$ and $B D C$,
$B C^{2}=9^{2}+7^{2}-2 \cdot 9 \cdot 7 \cdot \cos \beta=130-126 \cos \beta$ and

$$
B C^{2}=8^{2}+4^{2}-2 \cdot 8 \cdot 4 \cdot \cos \beta=80-64 \cos \beta
$$

From the equation $130-126 \cos \beta=80-64 \cos \beta$, we find that $\cos \beta=\frac{25}{31}$. Then $\sin \beta=\sqrt{1-\cos ^{2} \beta}=\frac{4 \sqrt{21}}{31}$.

Therefore,

$$
S_{\triangle \mathrm{BDC}}=\frac{1}{2} \cdot B D \cdot C D \cdot \sin \beta=\frac{1}{2} \cdot 8 \cdot 4 \cdot \frac{4 \sqrt{21}}{31}=\frac{64 \sqrt{21}}{31} .
$$

Thus,

$$
S_{\mathrm{ABCD}}=S_{\triangle \mathrm{ABD}}+S_{\Delta \mathrm{BDC}}=\frac{36 \sqrt{21}}{11}+\frac{64 \sqrt{21}}{31}=\frac{1820 \sqrt{21}}{341} .
$$

![](https://cdn.mathpix.com/cropped/2024_05_06_aa2960437c2efca8c53fg-39.jpg?height=506&width=535&top_left_y=816&top_left_x=767)

Answer

| 102413 Topics: | [ $\quad$ Law of Sines |
| :---: | :---: |

$\frac{1820 \sqrt{21}}{341}$.
## [

 The Law of Cosines [Area of a Triangle (using two sides and the included angle).] Quadrilateral $A B C D$ is inscribed in a circle. The lengths of the opposite sides $A B$ and $C D$ are 9 and 4, respectively, $A C=7, B D=8$. Find the area of quadrilateral $A B C D$.

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