Question

Using the technique in the model above, find the missing sides in this 30°-60°-90° right triangle. The hypotenuse is 12 and one of the shorter sides is 6. What are the lengths of the other two sides?

1) 4√3
2) 6√3
3) 8
4) 10

Answer 1

The lengths of the sides in a 30°-60°-90° triangle are proportional, with a ratio of 1:√3:2. For a triangle with a hypotenuse of 12, the shorter leg is 6 and the longer leg (opposite the 60° angle) is (2) 6√3. This is confirmed by the Pythagorean theorem.

Step-by-step explanation:

To find the missing sides of a 30°-60°-90° right triangle when the hypotenuse is 12 and one of the shorter sides is 6, we can utilize properties specific to this type of triangle. In a 30°-60°-90° triangle, the lengths of the sides are in the ratio 1:√3:2. Since the hypotenuse is the longest side, and is given as 12, this is twice the length of the shorter leg. If one of the shorter sides (the one opposite to the 30° angle) is 6, this is the shorter leg. Hence, the other shorter side, which is opposite the 60° angle, can be found by multiplying the shorter leg by √3.

Therefore, using the properties of 30°-60°-90° triangles:

• Shorter leg (opposite 30°): 6 (given)
• Longer leg (opposite 60°): 6√3
• Hypotenuse (opposite 90°): 12 (given)

To confirm, we can use the Pythagorean theorem:
6² + (6√3)² = c²

Calculating the longer leg:

6² + (6√3)²
= 36 + 36∗3
= 108

Since 108 is the square of the hypotenuse, we can see that the hypotenuse (12) matches the calculated hypotenuse:

√108 = 12