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 36 inches, and the length of the base is 12 inches. find the triangles perimeter. round to the nearest tenth of an inch.

Explanation:

Step1: Find half - base length

Since the altitude cuts the base into two equal segments and the base length is 12 inches, the length of each segment is $\frac{12}{2}=6$ inches.

Step2: Use Pythagorean theorem to find side length

Let the length of each of the equal sides of the isosceles triangle be $s$. In one of the right - triangles formed by the altitude, the two legs are 6 inches and 36 inches. By the Pythagorean theorem $a^{2}+b^{2}=c^{2}$, where $a = 6$, $b = 36$, and $c=s$. So $s=\sqrt{6^{2}+36^{2}}=\sqrt{36 + 1296}=\sqrt{1332}\approx36.5$.

Step3: Calculate the perimeter

The perimeter $P$ of an isosceles triangle with two equal sides of length $s$ and base of length $b$ is $P = 2s + b$. Substituting $s\approx36.5$ and $b = 12$, we get $P=2\times36.5+12=73 + 12=85.0$ inches.

Answer:

85.0 inches