If the hypotenuse of a 30-60-90 triangle is 8 cm, how long is the side opposite the 60-degree angle?(1 point)

The side opposite the 60-degree angle is cm.

1 answer

In a 30-60-90 triangle, the lengths of the sides are in the ratio \(1 : \sqrt{3} : 2\). Here, the hypotenuse is the longest side, which corresponds to the "2" in the ratio.

Given that the hypotenuse is 8 cm, you can find the side opposite the 60-degree angle using the ratio. This side corresponds to \(\sqrt{3}\) in the ratio.

To find the lengths of the sides:

  1. The length of the side opposite the 30-degree angle (shortest side) is \( \frac{1}{2} \times \text{hypotenuse} = \frac{1}{2} \times 8 = 4 \) cm.
  2. The length of the side opposite the 60-degree angle is \( \frac{\sqrt{3}}{2} \times \text{hypotenuse} = \frac{\sqrt{3}}{2} \times 8 = 4\sqrt{3} \).

Thus, the side opposite the 60-degree angle is \( 4\sqrt{3} \) cm, which is approximately 6.93 cm.

The exact answer is: The side opposite the 60-degree angle is \( 4\sqrt{3} \) cm.