Question

explain how to use the law of sines to find the perimeter of (\triangle pqr). then write an expression for the perimeter.
use the law of sines to get (qr). then use the triangle angle sum theorem to find (mangle q). then use the law of sines to find
use the law of sines to get (pq). then use the law of sines to find (mangle q). then use the law of sines to get (qr). the perimeter
an expression for the perimeter is (square+\frac{9sin62^{circ}}{sinsquare^{circ}}+\frac{9sinsquare^{circ}}{sinsquare^{circ}}) (type integers or decimals.)

Explanation:

Step1: Find \(m\angle Q\)

By the Triangle Angle - Sum Theorem, \(m\angle Q=180^{\circ}-(x^{\circ}+z^{\circ})\)

Step2: Apply the Law of Sines

The Law of Sines states that \(\frac{PQ}{\sin z^{\circ}}=\frac{QR}{\sin x^{\circ}}=\frac{PR}{\sin(180-(x + z))^{\circ}}=\frac{9}{\sin(180-(x + z))^{\circ}}\) (since \(PR = 9\)). So \(PQ=\frac{9\sin z^{\circ}}{\sin(180-(x + z))^{\circ}}\) and \(QR=\frac{9\sin x^{\circ}}{\sin(180-(x + z))^{\circ}}\)

Step3: Calculate the perimeter

The perimeter \(P\) of \(\triangle PQR\) is \(P = 9+PQ + QR\). Substituting the values of \(PQ\) and \(QR\) from the Law of Sines, we get \(P=9+\frac{9\sin z^{\circ}}{\sin(180-(x + z))^{\circ}}+\frac{9\sin x^{\circ}}{\sin(180-(x + z))^{\circ}}\)

Answer:

\(9+\frac{9\sin z^{\circ}}{\sin(180-(x + z))^{\circ}}+\frac{9\sin x^{\circ}}{\sin(180-(x + z))^{\circ}}\)