Question

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 9 inches, and the length of the base is 7 inches. find the triangles perimeter. round to the nearest tenth of an inch.

Explanation:

Step1: Analyze the isosceles triangle

In an isosceles triangle, the altitude to the base bisects the base. Given the base length is 7 inches, so each segment of the base (let's call it \( b \)) is \( \frac{7}{2}=3.5 \) inches. The altitude (let's call it \( h \)) is 9 inches.

Step2: Find the length of the equal sides

Using the Pythagorean theorem for one of the right - angled triangles formed (where the equal side of the isosceles triangle is the hypotenuse \( l \), one leg is the altitude \( h = 9 \) and the other leg is \( b=3.5 \)). The Pythagorean theorem is \( l=\sqrt{h^{2}+b^{2}} \). Substitute \( h = 9 \) and \( b = 3.5 \) into the formula: \( l=\sqrt{9^{2}+3.5^{2}}=\sqrt{81 + 12.25}=\sqrt{93.25}\approx9.656 \) inches.

Step3: Calculate the perimeter

The perimeter \( P \) of the isosceles triangle is the sum of the lengths of the two equal sides and the base. So \( P=2l + 7 \). Substitute \( l\approx9.656 \) into the formula: \( P=2\times9.656+7=19.312 + 7=26.312\approx26.3 \) inches.

Answer:

The perimeter of the isosceles triangle is approximately 26.3 inches.