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 18 inches, and the length of the base is 15 inches. find the triangles perimeter. round to the nearest tenth of an inch.

Explanation:

Step1: Find the length of the equal sides

Since the altitude of an isosceles triangle divides the base into two equal segments, each segment of the base is \(b=\frac{15}{2} = 7.5\) inches.
Using the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(a = 18\) inches (altitude) and \(b=7.5\) inches (half - base), we can find the length of the equal sides \(c\).

$$c=\sqrt{18^{2}+7.5^{2}}=\sqrt{324 + 56.25}=\sqrt{380.25}=19.5$$

Step2: Calculate the perimeter

The perimeter \(P\) of a triangle is \(P=a + b+ c\). In an isosceles triangle with base \(b = 15\) and equal sides \(c = 19.5\), we have \(P=19.5+19.5 + 15\)

$$P=19.5\times2+15=39 + 15=54$$

Answer:

The perimeter of the triangle is \(54.0\) inches.