Question

3. Priya says, "I know everything about a right triangle with a 30 degree angle and a hypotenuse with length 1 cm. Here look." The other angle is 60 degrees. The leg adjacent to the 30 degree angle 0.866 cm long. The side opposite the 30 degree angle is 0.5 cm long. Han asks, "What would happen if a right triangle with a 30 degree angle has a hypotenuse that is 2 cm instead?" Help them find the missing angles and side lengths in the new triangle. Explain or show your reasoning.

Answer 1

1) The measure of the third angle is 60°

2) The length of the opposite side to the angle 30° is 1 cm

3) The length of the opposite side to the third angle 60° is approximately 1.732 cm

Explanation:

1) The given parameters are;

The known angle of a right triangle = 30°

The length of the hypotenuse = 2 cm

Given that the triangle = A right triangle, with one of the interior angles = 90°, we have;

The measure of the third angle = 180° - 90° - 30° = 60°

The measure of the third angle = 60°

2) We have that sin(30°) = 0.5 = (The length of the opposite side to the angle 30°)/(The length of the hypotenuse side)

0.5 = (The length of the opposite side to the angle 30°)/(2 cm)

∴ The length of the opposite side to the angle 30° = 2 cm × 0.5 = 1 cm

The length of the opposite side to the angle 30° = 1 cm

3) Similarly, the length of the opposite side to the third angle 60° = sin(60°) × (The length of the hypotenuse side)

∴ The length of the opposite side to the third angle 60° = (√3)/2 × 2 cm = √3 cm

The length of the opposite side to the third angle 60° = √3 cm ≈ 1.732 cm