An equilateral triangle has equal sides of 10 in. Find its perpendicular height.

Step 1: Understand the properties of an equilateral triangle. An equilateral triangle has all sides equal and all angles equal to 60 degrees.

Step 2: Draw a line from one angle to the midpoint of the opposite side. This line is the height (h) of the triangle and it divides the triangle into two 30-60-90 right triangles.

Step 3: In a 30-60-90 triangle, the sides are in the ratio 1:√3:2. The longest side (hypotenuse) is twice the shortest side (opposite the 30 degree angle), and the remaining side (opposite the 60 degree angle) is √3 times the shortest side.

Step 4: In this case, the hypotenuse of the right triangle is the side of the equilateral triangle, which is 10 inches. So, the height of the triangle is the side opposite the 60 degree angle, which is √3 times the shortest side.

Step 5: The shortest side of the right triangle is half the length of the hypotenuse, which is 10/2 = 5 inches.

Step 6: Therefore, the height of the equilateral triangle is 5√3 inches.