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 13 inches, and the length of the base is 8 inches. Find the triangle’s perimeter. Round to the nearest tenth of an inch.
Let's call the length of each leg of the isosceles triangle "x". Since the altitude splits the triangle into two congruent right triangles, we know that the length of one leg of each right triangle is 4 (half of the base, which is 8). Using the Pythagorean theorem, we can solve for the length of the other leg:
x^2 = 13^2 + 4^2
x^2 = 185
x ≈ 13.6
Now we can find the perimeter of the whole triangle:
P = 2x + 8
P = 2(13.6) + 8
P ≈ 35.2
Rounding to the nearest tenth of an inch, the perimeter is 35.2 inches.
To find the perimeter of the triangle, we need to find the lengths of the two congruent triangles created by the altitude.
1. Let's start by labeling the triangle. Let the base of the isosceles triangle be AB and the vertex be C. Let the midpoint of the base be D, where the altitude intersects the base.
2. Since the altitude creates a right angle, we can use the Pythagorean theorem to find the lengths of the congruent triangles. Let's call the length of one of the congruent sides x.
3. Applying the Pythagorean theorem, we have:
x^2 + 6^2 = 13^2
x^2 + 36 = 169
x^2 = 169 - 36
x^2 = 133
x = √133 (taking the square root of both sides)
4. Now that we have the length of one side of the congruent triangles, we can find the perimeter of the whole triangle. The perimeter is the sum of all three sides.
5. The length of the base AB is 8 inches, so the length of the other congruent side is also 8 inches.
6. The length of the hypotenuse of the right triangles (opposite the right angle) is the sum of the two congruent sides. Therefore, the hypotenuse is 2x.
7. The perimeter of the triangle is given by: AB + AC + BC
Since AB = 8 inches, the perimeter is 8 + 2x + 2x (since AC and BC are congruent)
Hence, the perimeter is: 8 + 2x + 2x = 8 + 4x
8. Substituting x = √133 into the equation, we have:
Perimeter = 8 + 4(√133)
Perimeter ≈ 8 + 4(11.53)
Perimeter ≈ 8 + 46.12
Perimeter ≈ 54.12 inches
Therefore, the perimeter of the triangle is approximately 54.12 inches.