7. Adam has a triangular field $A B C$ with $A B=5, B C=8$ and $C A=11$. He intends to separate the field into two parts by building a straight fence from $A$ to a point $D$ on side $B C$ such that $A D$ bisects $\angle B A C$. Find the area of the part of the field $A B D$.
(A) $\frac{4 \sqrt{21}}{11}$;
(B) $\frac{4 \sqrt{21}}{5}$;
(C) $\frac{5 \sqrt{21}}{11}$;
(D) $\frac{5 \sqrt{21}}{4}$;
(E) None of the above.
7. Answer: (D)
By Heron's formula, $S_{\triangle A B C}=\sqrt{12(12-5)(12-8)(12-11)}=4 \sqrt{21}$.
$$
\frac{S_{\triangle A B D}}{S_{\triangle A C D}}=\frac{\frac{1}{2} \times A B \times A D \times \sin \alpha}{\frac{1}{2} \times A C \times A D \times \sin \alpha}=\frac{A B}{A C}=\frac{5}{11}
$$

Then
$$
S_{\triangle A B D}=\frac{5}{5+11} \times 4 \sqrt{21}=\frac{5 \sqrt{21}}{4}
$$