Exam

Olympiad workbook

Subject

IOQM

Chapter

Properties of Triangles

Olympiad workbook · IOQM · Properties of Triangles

In a triangle ABC , it is known that A=100^ and AB=AC . The...

In a triangle ABC , it is known that $\angle \mathrm{A}=100^{\circ}$ and $\mathrm{AB}=\mathrm{AC}$. The internal angle bisector BD has length 20 units. Find the length of BC to the nearest integer, given that $\sin 10^{\circ} \approx 0.174$.

Step-by-step solution

$\triangle \mathrm{BCD}$, by sine rule
$$\begin{aligned} & \frac{B C}{120^{\circ}}=\frac{B D}{\sin 40^{\circ}}=\frac{20}{\sin 40^{\circ}} \\ & \therefore B C=\frac{10 \sqrt{3}}{\sin 40^{\circ}} \end{aligned}$$ Given, $\cos 80^{\circ}\left(\sin 10^{\circ}\right) \approx 0.174$ $\sin 40^{\circ} \approx \sqrt{\frac{1-\cos 80^{\circ}}{2}} \approx 0.643$ $\therefore$ Nearest integer to $\mathrm{BC}=27$