Question

in △abc, m∠cab = 60°, $overline{ad}$ is the angle - bisector of ∠bac with d ∈ $overline{bc}$ and ad = 8 ft. find the distances from point d to the sides of the triangle.

Explanation:

Step1: Recall angle - bisector property

The distance from a point on an angle - bisector to the two sides of the angle is equal. Let the distance from D to AB and AC be h.

Step2: Use trigonometry

Since $\angle CAB = 60^{\circ}$ and AD is the angle - bisector, $\angle CAD=\angle BAD = 30^{\circ}$. In right - triangle formed by the perpendicular from D to a side (say AC) and AD, $\sin\angle CAD=\frac{h}{AD}$.

Step3: Substitute values

Given AD = 8 ft and $\sin30^{\circ}=\frac{1}{2}$, we have $\frac{h}{8}=\frac{1}{2}$, so $h = 4$ ft.

Answer:

4 ft