Question

what is ac? use both the distance formula and the law of sines. how do the values compare? explain.
(5ive an integer or decimal rounded to four decimal places as needed.)
using the law of sines. ac =
(type an integer or decimal rounded to four decimal places as needed.)

Explanation:

Step1: Use the Distance Formula

The coordinates of \(A(-2,1)\) and \(B(2,1)\). Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), for \(AB\): \(x_1=-2,y_1 = 1,x_2=2,y_2 = 1\). Then \(AB=\sqrt{(2-(-2))^2+(1 - 1)^2}=\sqrt{(4)^2+0}=4\).

Step2: Use the Law of Sines

In \(\triangle ABC\), by the Law of Sines \(\frac{AC}{\sin B}=\frac{AB}{\sin C}\). First, find \(\angle C=180^{\circ}-63.43^{\circ}-71.47^{\circ}=45.1^{\circ}\). Given \(AB = 4\), \(B = 71.47^{\circ}\), \(C=45.1^{\circ}\). Then \(AC=\frac{AB\times\sin B}{\sin C}\). Substitute the values: \(AC=\frac{4\times\sin(71.47^{\circ})}{\sin(45.1^{\circ})}\). Since \(\sin(71.47^{\circ})\approx0.9487\) and \(\sin(45.1^{\circ})\approx0.7080\), \(AC=\frac{4\times0.9487}{0.7080}\approx5.31\).

Answer:

\(AC\approx5.31\)