Question

question 2 of 10
a certain triangle has a 30° angle and a 60° angle. which must be a true statement about the triangle?
a. the second - longest side is √3 times as long as the shortest side.
b. the longest side is twice as long as the second - longest side.
c. the longest side is √3 times as long as the shortest side.
d. two sides of the triangle have the same length.

Explanation:

Step1: Find the third angle

The sum of angles in a triangle is \(180^\circ\). Given angles are \(30^\circ\) and \(60^\circ\), so the third angle is \(180 - 30 - 60 = 90^\circ\). So it's a 30 - 60 - 90 right triangle.

Step2: Recall side ratios in 30 - 60 - 90 triangle

In a 30 - 60 - 90 triangle, the side ratios (opposite 30°, 60°, 90°) are \(1:\sqrt{3}:2\). Let the side opposite \(30^\circ\) (shortest side) be \(x\), opposite \(60^\circ\) (second - longest) be \(x\sqrt{3}\), opposite \(90^\circ\) (longest) be \(2x\).

Step3: Analyze each option

• Option A: Second - longest side (\(x\sqrt{3}\)) is \(\sqrt{3}\) times shortest side (\(x\)). This is true.
• Option B: Longest side (\(2x\)) and second - longest (\(x\sqrt{3}\)): \(\frac{2x}{x\sqrt{3}}=\frac{2}{\sqrt{3}}

eq2\). So false.

• Option C: Longest side (\(2x\)) and shortest (\(x\)): \(\frac{2x}{x} = 2

eq\sqrt{3}\). False.

• Option D: Sides are \(x,x\sqrt{3},2x\), no two sides equal. False.

Answer:

A. The second - longest side is \(\sqrt{3}\) times as long as the shortest side.