Question

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

Explanation:

Step1: Recall area formula for isosceles triangle

The area formula for a triangle with two - equal sides \(a\) and the included angle \(\theta\) is \(A=\frac{1}{2}a^{2}\sin\theta\).

Step2: Substitute given values into formula

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

Step3: Solve for \(a^{2}\)

First, simplify the right - hand side of the equation: \(22=\frac{a^{2}}{4}\). Then, multiply both sides by 4 to isolate \(a^{2}\), so \(a^{2}=88\).

Step4: Solve for \(a\)

Take the square root of both sides: \(a=\sqrt{88}\approx 9.4\space cm\).

Answer:

9.4