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. answer: ft

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 point D to AB and AC be h.

Step2: Use trigonometry in right - triangles

Since AD is the angle bisector of ∠BAC and ∠BAC = 60°, then ∠BAD=∠DAC = 30°. Consider a right - triangle formed by the perpendicular from D to one of the sides (say AB) and AD. In a right - triangle, if the angle is 30° and the hypotenuse is AD = 8 ft, and we know that in a right - triangle with an angle θ, sinθ=\(\frac{\text{opposite}}{\text{hypotenuse}}\). Here, if θ = 30° and the hypotenuse is AD, and the opposite side is the distance h from D to the side of the triangle.
We have sin30°=\(\frac{h}{AD}\).

Step3: Substitute the value of AD

Since sin30°=\(\frac{1}{2}\) and AD = 8 ft, we substitute into the equation \(\frac{1}{2}=\frac{h}{8}\).

Step4: Solve for h

Cross - multiply to get h = 4 ft.

Answer:

4