Question

4 practice 4 (from unit 4, lesson 3)
triangle abc, shown here, is equilateral.
image of triangle abc with ad perpendicular to cb, ac = 8, angle acd congruent to angle abd
a. what is the value of x?
type your answer in the box.
x = \boxed{ } units
b. what is the measure of angle b?
type your answer in the box.
\boxed{ } ^\circ
how did i do?

Explanation:

Part a

Step1: Use Pythagorean theorem

In right - triangle \(ACD\), \(AC = 8\) (side of equilateral triangle), \(CD=\frac{CB}{2}\) (since \(AD\perp CB\) and \(\triangle ABC\) is equilateral). In an equilateral triangle \(CB = AC=8\), so \(CD = 4\).
By Pythagorean theorem \(x=\sqrt{AC^{2}-CD^{2}}\)

Step2: Calculate the value of \(x\)

Substitute \(AC = 8\) and \(CD = 4\) into the formula:

$$x=\sqrt{8^{2}-4^{2}}=\sqrt{64 - 16}=\sqrt{48}=4\sqrt{3}$$

Part b

Step1: Recall the property of an equilateral triangle

In an equilateral triangle, all angles are equal. The sum of interior angles of a triangle is \(180^{\circ}\). Let the measure of each angle be \(\theta\).
We use the formula \(\theta=\frac{180^{\circ}}{n}\), where \(n = 3\) (number of angles in a triangle)

Step2: Calculate the measure of angle \(B\)

\(\theta=\frac{180^{\circ}}{3}=60^{\circ}\)

Answer:

a. \(x = 4\sqrt{3}\)
b. \(60\)