Final answer:
The height of an equiangular triangle can be found using the Pythagorean theorem. In this case, the height is approximately 8.7 inches.
Step-by-step explanation:
An equiangular triangle is a triangle where all three angles are equal. Since it is equiangular, we know that the height of the triangle will bisect the base and form two congruent right triangles. Using the Pythagorean theorem, we can find the height. Let's call the height h and the base b. The Pythagorean theorem states that the sum of the squares of the two legs of a right triangle is equal to the square of the hypotenuse. In this case, the hypotenuse is the side of length 10 inches, and one leg is the height h. The other leg is the half of the base, which is b/2. Therefore, we have the equation h^2 + (b/2)^2 = 10^2. Since the triangle is equiangular, all three sides are congruent. So, we have h^2 + (b/2)^2 = b^2. Plugging in the value of one side, 10 inches, we can solve for the height:
h^2 + (10/2)^2 = 10^2
h^2 + 25 = 100
h^2 = 100 - 25
h^2 = 75
h = √75
h ≈ 8.7 inches
The height of the equiangular triangle is approximately 8.7 inches.