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: Recall equilateral - triangle properties

In equilateral triangle \(ABC\), \(AD\) is the perpendicular bisector of \(BC\). Let the side - length of \(\triangle ABC\) be \(s\). Then \(BD = DC\) and \(\angle BAD=30^{\circ}\), \(\angle B = 60^{\circ}\).

Step2: Use 30 - 60 - 90 triangle ratios

In right - triangle \(ABD\), if \(AD = 18\) inches, and the ratio of sides in a \(30 - 60 - 90\) triangle is \(1:\sqrt{3}:2\). If the side opposite the \(30^{\circ}\) angle is \(x\), the side opposite the \(60^{\circ}\) angle is \(x\sqrt{3}\), and the hypotenuse is \(2x\). Here \(AD\) (opposite \(60^{\circ}\)) is \(18\), so \(BD=\frac{18}{\sqrt{3}} = 6\sqrt{3}\) inches, and \(AB = 12\sqrt{3}\) inches. Also, \(BC=AC = AB = 12\sqrt{3}\) inches.

Answer:

AC = 12√3 in., DC = 6√3 in.