Question

which of the following are true statements about a 30 - 60 - 90 triangle? check all that apply. a. the longer leg is √3 times as long as the shorter leg. b. the hypotenuse is √3 times as long as the longer leg. c. the hypotenuse is twice as long as the longer leg. d. the hypotenuse is twice as long as the shorter leg. e. the hypotenuse is √3 times as long as the shorter leg. f. the longer leg is twice as long as the shorter leg.

Explanation:

Step1: Recall 30 - 60 - 90 triangle ratios

Let the shorter leg (opposite the 30 - degree angle) be \(a\). The longer leg (opposite the 60 - degree angle) is \(a\sqrt{3}\), and the hypotenuse (opposite the 90 - degree angle) is \(2a\).

Step2: Analyze Option A

If the shorter leg is \(a\) and the longer leg is \(b\), then \(b = a\sqrt{3}\), so the longer leg is \(\sqrt{3}\) times as long as the shorter leg. Option A is true.

Step3: Analyze Option B

The hypotenuse \(c = 2a\) and the longer leg \(b=a\sqrt{3}\). Since \(c
eq b\sqrt{3}\), Option B is false.

Step4: Analyze Option C

The hypotenuse \(c = 2a\) and the longer leg \(b=a\sqrt{3}\). Since \(c
eq 2b\), Option C is false.

Step5: Analyze Option D

The hypotenuse \(c = 2a\) where \(a\) is the shorter leg, so the hypotenuse is twice as long as the shorter leg. Option D is true.

Step6: Analyze Option E

The hypotenuse \(c = 2a\) and the shorter leg is \(a\). Since \(c
eq a\sqrt{3}\), Option E is false.

Step7: Analyze Option F

The longer leg \(b=a\sqrt{3}\) and the shorter leg is \(a\). Since \(b
eq 2a\), Option F is false.

Answer:

A. The longer leg is \(\sqrt{3}\) times as long as the shorter leg.
D. The hypotenuse is twice as long as the shorter leg.