Question

triangle abc is an equilateral triangle. segment ad measures 18 inches. which statements about the diagram are correct? check all that apply. bd = 9 in. ab = 36 in. dc = 6√3 in. ac = 12√3 in. bc = 18√3 in.

Explanation:

Step1: Identify triangle properties

ABC is equilateral, so $AB=BC=AC$, and AD is a perpendicular bisector, meaning $BD=DC$, and $\triangle ABD$ is a 30-60-90 right triangle.

Step2: Relate AD to side length

In 30-60-90 triangle, height $AD = \frac{\sqrt{3}}{2} \times AB$. Solve for $AB$:
$$AB = \frac{2 \times AD}{\sqrt{3}} = \frac{2 \times 18}{\sqrt{3}} = 12\sqrt{3}$$
Since $AB=AC=BC$, $AC=12\sqrt{3}$ in, $BC=12\sqrt{3}$ in.

Step3: Calculate BD/DC

$BD = \frac{1}{2} \times AB = \frac{1}{2} \times 12\sqrt{3} = 6\sqrt{3}$ in, so $DC=6\sqrt{3}$ in.

Answer:

DC = $6\sqrt{3}$ in, AC = $12\sqrt{3}$ in