triangle abc is an equilateral triangle. segment ad measures 18 inches. which statements about the diagram…

Answer:

• A. BD = 9 in.
• D. AC = 12√3 in.
• E. BC = 12√3 in.

Explanation:

Step1: Recall equilateral - triangle properties

In an equilateral triangle (ABC) with (AD\perp BC), (BD = DC=\frac{1}{2}BC) and (\angle B = 60^{\circ}), (\angle BAD=30^{\circ}).

Step2: Use 30 - 60 - 90 triangle ratios

In right - triangle (ABD), if the side opposite the (30^{\circ}) angle is (x), the side opposite the (60^{\circ}) angle ((AD)) is (x\sqrt{3}), and the hypotenuse ((AB)) is (2x). Given (AD = 18) inches. Let (BD=x), then (AD=x\sqrt{3}). Since (AD = 18), we have (x\sqrt{3}=18), so (x = BD=DC = 6\sqrt{3}) inches.

Step3: Calculate side lengths of the equilateral triangle

The side length of the equilateral triangle (AB = AC=BC = 2x=12\sqrt{3}) inches.

• For (BD): Since (AD) is the perpendicular bisector of (BC) in equilateral (\triangle ABC), and using the 30 - 60 - 90 triangle relationship in (\triangle ABD) where (AD = 18) and (\angle BAD = 30^{\circ}), (BD=\frac{18}{\sqrt{3}}=6\sqrt{3}\neq9), so the statement (BD = 9) in. is incorrect.
• For (AB): Using the 30 - 60 - 90 triangle relationship in (\triangle ABD), if (BD=x) and (AD = 18=x\sqrt{3}), then (x = 6\sqrt{3}) and (AB = 2x=12\sqrt{3}\neq36), so the statement (AB = 36) in. is incorrect.
• For (DC): Since (AD) is the perpendicular bisector of (BC) in equilateral (\triangle ABC), (DC = BD). From (AD = 18) and the 30 - 60 - 90 triangle relationship ((AD=\sqrt{3}BD)), (BD=DC = 6\sqrt{3}) inches, this statement is correct.
• For (AC): In right - triangle (ADC), (AD = 18), (DC = 6\sqrt{3}). By the Pythagorean theorem (AC=\sqrt{AD^{2}+DC^{2}}=\sqrt{18^{2}+(6\sqrt{3})^{2}}=\sqrt{324 + 108}=\sqrt{432}=12\sqrt{3}) inches, this statement is correct.
• For (BC): Since (BC = 2BD) and (BD = 6\sqrt{3}), (BC=12\sqrt{3}) inches, this statement is correct.