Find all the sides in the other angle of the following triangle.

Question

The triangle is a 30, 60, 90 triangle with the shortest side being 6 and the long side being 12. What is the value of the other short side. Write the exact value no decimals.

Answer 1

The conditions describe a 30-60-90 right triangle. Because we are given the longest side, 12, and we know that it is opposite the largest angle , 90, and we are given the shortest side , 6, and we know it is opposite the smallest angle, 30, we see that we can apply the Pythagorean Theorem, to find the missing side that is opposite the angle, 60.

Using the side-hypotenuse-side form of the Pythagorean Theorem:

1. b² ₌ c² - a²

we can substitute and solve:

b² = 12² - 6² = 144 - 36 = 108

taking the square root:

b = √ 108 = √ 2².3².3 = √ 2². √ 3².√3 = 2.3.√3 = 6√3 is the side opposite the 60 degree angle.

the answer is in closed form as requested.

Answer 2

Draw and label a diagram!

If the short side (opposite the 30° angle) is one-half the hypotenuse.

Therefore the hypotenuse is 12.

The long side cannot be 12.