An altitude is drawn from the vertex of an isosceles triangle, forming a right angle and two congruent triangles. As a result, the altitude cuts the base into two equal segments. The length of the altitude is 36 inches, and the length of the base is 12 inches. Find the triangle’s perimeter. Round to the nearest tenth of an inch.

To find the perimeter of the isosceles triangle, we first need to find the length of the two equal sides. Since the altitude forms a right angle with the base and cuts it into two equal segments, we can use the Pythagorean theorem to find the length of the equal sides.

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

In this case, the length of the base of each of the two right triangles formed by the altitude is 12/2 = 6 inches (since the altitude cuts the base into two equal segments), and the length of the altitude is 36 inches.

So, we can set up the equation as follows:

a² = b² + c²

where a is the length of the hypotenuse (the side of the isosceles triangle we're trying to find), b is the length of the altitude, and c is half the length of the base.

Substituting the given values, we get:

a² = 36² + 6² a² = 1296 + 36 a² = 1332

Taking the square root of both sides to solve for a, we get:

a = √1332 ≈ 36.5 inches

So, the length of each of the equal sides of the isosceles triangle is approximately 36.5 inches.

Finally, to find the perimeter of the triangle, we add up the lengths of all its sides:

Perimeter = 2a + base Perimeter = 2(36.5) + 12 Perimeter = 73 + 12 Perimeter = 85 inches

So, the perimeter of the triangle is approximately 85 inches, rounded to the nearest tenth of an inch.