Question

question 7 of 10
which of the following are true statements about a 30 - 60 - 90 triangle?
check all that apply.
a. the hypotenuse is twice as long as the longer leg.
b. the hypotenuse is twice as long as the shorter leg.
c. the hypotenuse is \\( \sqrt { 3 } \\) times as long as the longer leg.
d. the longer leg is \\( \sqrt { 3 } \\) times as long as the shorter leg.
e. the hypotenuse is \\( \sqrt { 3 } \\) times as long as the shorter leg.
f. the longer leg is twice as long as the shorter leg

Explanation:

To determine the true statements about a 30 - 60 - 90 triangle, we recall the side ratios of a 30 - 60 - 90 triangle. In a 30 - 60 - 90 triangle, if the length of the shorter leg (opposite the 30° angle) is \(x\), then:

• The hypotenuse is \(2x\) (so the hypotenuse is twice the shorter leg).
• The longer leg (opposite the 60° angle) is \(x\sqrt{3}\) (so the longer leg is \(\sqrt{3}\) times the shorter leg, and the hypotenuse \(2x=\frac{2x\sqrt{3}}{\sqrt{3}}=\frac{2}{\sqrt{3}}\times(x\sqrt{3})\), which means the hypotenuse is \(\frac{2}{\sqrt{3}}\) times the longer leg, or equivalently, the longer leg is \(\frac{\sqrt{3}}{2}\) times the hypotenuse).

Now let's analyze each option:

• Option A: The hypotenuse is \(2x\) and the longer leg is \(x\sqrt{3}\). \(2x

eq2\times(x\sqrt{3})\), so A is false.

• Option B: The hypotenuse is \(2x\) and the shorter leg is \(x\), so \(2x = 2\times x\), B is true.
• Option C: The hypotenuse is \(2x\) and the longer leg is \(x\sqrt{3}\). \(2x=\frac{2}{\sqrt{3}}\times(x\sqrt{3})\approx1.1547\times(x\sqrt{3})

eq\sqrt{3}\times(x\sqrt{3})\), so C is false.

• Option D: The longer leg is \(x\sqrt{3}\) and the shorter leg is \(x\), so \(x\sqrt{3}=\sqrt{3}\times x\), D is true.
• Option E: The hypotenuse is \(2x\) and the shorter leg is \(x\), \(2x

eq\sqrt{3}\times x\), so E is false.

• Option F: The longer leg is \(x\sqrt{3}\) and the shorter leg is \(x\), \(x\sqrt{3}

eq2\times x\), so F is false.

Answer:

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