An equiangular triangle has one side of length six inches. What is the height of the triangle, drawn from that side, to the nearest tenth of an inch?

Question

An equiangular triangle has one side of length six inches. What is the height of the triangle, drawn from that side, to the nearest tenth of an inch?

Answer 1

An equiangular triangle has all angles and all sides with equal measures. Since a triangle is 180 degrees total, we can divide by 3 angles, which gives us 60 degrees in each corner.

The sides are all 6 inches.

To find the height of the triangle, cut the triangle in half. This gives us the 30-60-90 degree triangle. the base of the smaller triangle is 3 inches and the hypothenuse is 6 inches.

There are two options to solve for the height:

Option 1: Use the Pythagorean Theorem a^2+b^2=c^2 where a=3 and c=6.

Then we have 3^2+b^2=6^2, which is 9+b^2=36 and then subtract 9 from both sides to get b^2=27. Take the square root of both sides (you do not need +/- in this case since negative length does not make sense). So the answer is sqrt(27) which is rounded to 5.2 inches.

Option 2: The ratio of the sides for the 30-60-90 degree triangle is 1x (shorter side) to 2x (longer side) to x*sqrt2 (hypothenuse). In this problem, the shortest side is 3, so 1x=3, which means x=3. Therefore, x*sqrt2 is 3sqrt2, which is rounded to 5.2 inches.