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 for each angle.

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 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.

So the answer is 5.2 inches