Question

1. triangle abc, shown here, is equilateral.

a. what is the value of x?
b. what is the measure of angle b?
∠acd ≅ ∠abd
( overline{ad} perp overline{cb} )

Explanation:

Step1: Find half of CB

Since ABC is equilateral, $CB = 8$. Half of $CB$ is $\frac{8}{2}=4$.

Step2: Solve for x via Pythagoras

In right triangle ABD, use $AB^2 = AD^2 + BD^2$.
$$8^2 = x^2 + 4^2$$
$$64 = x^2 + 16$$
$$x^2 = 64 - 16 = 48$$
$$x = \sqrt{48} = 4\sqrt{3} \approx 6.93$$

Step3: Identify angle B measure

All angles in equilateral triangle are equal. Sum of triangle angles is $180^\circ$, so each angle is $\frac{180^\circ}{3}=60^\circ$.

Answer:

a. $4\sqrt{3}$ (or approximately 6.93)
b. $60^\circ$