Question

an isosceles triangle has an area of 22 cm², and the angle between the two equal sides is 5π/6. what is the length of the two equal sides? (round your answer to one decimal place.)

Explanation:

Step1: Recall the area formula for an isosceles triangle

The area formula for a triangle with two - side lengths \(a\) and \(b\) and included angle \(\theta\) is \(A=\frac{1}{2}ab\sin\theta\). In an isosceles triangle with equal side lengths \(x\), \(a = b=x\), so \(A=\frac{1}{2}x^{2}\sin\theta\).

Step2: Substitute the given values into the formula

We know that \(A = 22\mathrm{cm}^{2}\) and \(\theta=\frac{5\pi}{6}\), and \(\sin(\frac{5\pi}{6})=\frac{1}{2}\). Substituting into \(A=\frac{1}{2}x^{2}\sin\theta\), we get \(22=\frac{1}{2}x^{2}\times\frac{1}{2}\).

Step3: Solve for \(x\)

First, simplify the equation \(22=\frac{1}{4}x^{2}\). Multiply both sides of the equation by \(4\) to get \(x^{2}=88\). Then take the square - root of both sides: \(x=\sqrt{88}\approx9.4\) (rounded to one decimal place).

Answer:

\(9.4\)