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: Properties of equilateral triangle

In an equilateral triangle \(ABC\), \(AD\perp BC\) and \(BD = DC\). Let \(BD=x\), \(AB = 2x\) (by the property of \(30 - 60-90\) triangle: in \(\triangle ABD\), \(\angle B = 60^{\circ}\), \(\angle ADB=90^{\circ}\), \(\angle BAD = 30^{\circ}\), and the sides are in the ratio \(1:\sqrt{3}:2\)). Using the Pythagorean theorem \(AB^{2}=BD^{2}+AD^{2}\).

Step2: Substitute values

Since \(AD = 18\) and \(AB = 2x\), \(BD=x\), we have \((2x)^{2}=x^{2}+18^{2}\). Expanding gives \(4x^{2}-x^{2}=324\), so \(3x^{2}=324\), then \(x^{2}=108\), \(x = 6\sqrt{3}\).

Step3: Calculate side lengths

\(BD=DC = 6\sqrt{3}\) (because \(BD = DC\) in equilateral triangle \(ABC\) with \(AD\) as altitude). \(AB=AC=BC = 2x=12\sqrt{3}\).

Answer:

\(DC = 6\sqrt{3}\text{ in}\), \(AC = 12\sqrt{3}\text{ in}\)