Question

question 4 of 28 what is the length of the hypotenuse in the 30 - 60 - 90 triangle shown below? a. $6sqrt{3}$ b. 12 c. $6sqrt{2}$ d. 3

Explanation:

Step1: Recall 30 - 60 - 90 triangle ratio

In a 30 - 60 - 90 triangle, if the side opposite the 30° angle is $x$, the side opposite the 60° angle is $x\sqrt{3}$ and the hypotenuse is $2x$.

Step2: Identify the side length

The side given is opposite the 60° angle and its length is 6. Let the side opposite the 30° angle be $x$. Since the side opposite the 60° angle is $x\sqrt{3}=6$, we can solve for $x$. We get $x = \frac{6}{\sqrt{3}}=\frac{6\sqrt{3}}{3}=2\sqrt{3}$.

Step3: Calculate the hypotenuse

The hypotenuse $c$ of a 30 - 60 - 90 triangle is $2x$. Substituting $x = 2\sqrt{3}$, we have $c = 2\times2\sqrt{3}=4\sqrt{3}$. Another way is to use the direct relationship with the side opposite 60°. If the side opposite 60° is $a = 6$, and we know $\sin60^{\circ}=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{\sqrt{3}}{2}$. Let the hypotenuse be $h$. Then $\frac{6}{h}=\frac{\sqrt{3}}{2}$, and $h=\frac{12}{\sqrt{3}} = 4\sqrt{3}$. But if we assume the side of length 6 is the shorter leg (opposite 30°), then the hypotenuse is $2\times6=12$. In a 30 - 60 - 90 triangle, the side opposite 30° is the shortest side. Since the side given is likely the shorter side (as no other information suggests otherwise), using the ratio of sides in a 30 - 60 - 90 triangle where hypotenuse = 2×(side opposite 30°), if the side opposite 30° is 6, the hypotenuse is 12.

Answer:

B. 12