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: Find the length of half - base

Since the altitude cuts the base into two equal segments and the base length \(b = 7\) inches, the length of half - base \(x=\frac{b}{2}=\frac{7}{2}=3.5\) inches.

Step2: Use the Pythagorean theorem to find the length of the equal sides

Let the length of the equal sides of the isosceles triangle be \(l\). According to the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(a = 3.5\) (half - base), \(b = 9\) (altitude) and \(c=l\) (equal side of the isosceles triangle).

$$l=\sqrt{3.5^{2}+9^{2}}=\sqrt{12.25 + 81}=\sqrt{93.25}\approx9.657$$

Step3: Calculate the perimeter of the isosceles triangle

The perimeter \(P\) of an isosceles triangle with base \(b = 7\) and equal sides \(l\approx9.657\) is \(P = 2l + b\).

$$P=2\times9.657+7=19.314 + 7=26.314\approx26.3$$

Answer:

\(26.3\) inches