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 11 inches, and the length of the base is 6 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 splits the base 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. We can write this as: a² + b² = c²

In this case, the length of the altitude is the height of the right triangles (11 inches), and half the length of the base is the base of the right triangles (3 inches). We're trying to find the hypotenuse of these right triangles, which is also the length of the equal sides of the isosceles triangle.

So, we can set up the equation as follows: 11² + 3² = c²

Solving for c, we get: c = sqrt(121 + 9) = sqrt(130) ≈ 11.4 inches

So, the length of the equal sides of the isosceles triangle is approximately 11.4 inches.

Finally, to find the perimeter of the isosceles triangle, we add up the lengths of all its sides: 11.4 inches + 11.4 inches + 6 inches = 28.8 inches

So, the perimeter of the triangle is approximately 28.8 inches, rounded to the nearest tenth of an inch.