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: Analyze the properties of an equilateral triangle

In an equilateral triangle \(ABC\), \(AD\perp BC\), so \(BD = DC=\frac{1}{2}BC\) and \(\angle BAD=\angle CAD = 30^{\circ}\). Let \(BD=x\), then \(AB = 2x\) (by the property of a \(30 - 60-90\) triangle: in a right - angled triangle with angles \(30^{\circ}-60^{\circ}-90^{\circ}\), the sides are in the ratio \(1:\sqrt{3}:2\)).
Using the Pythagorean theorem in \(\triangle ABD\): \(AB^{2}=BD^{2}+AD^{2}\). Since \(AB = 2x\) and \(AD = 18\), we have \((2x)^{2}=x^{2}+18^{2}\).

Step2: Solve the equation from the Pythagorean theorem

Expand \((2x)^{2}=x^{2}+18^{2}\) to get \(4x^{2}-x^{2}=324\), so \(3x^{2}=324\), then \(x^{2}=108\), \(x = 6\sqrt{3}\).
So \(BD=DC = 6\sqrt{3}\) inches, \(AB=AC=BC = 12\sqrt{3}\) inches.

Answer:

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