Question

question 4 of 25 calculate the length a to two decimal places.

Explanation:

Step1: Apply the Law of Cosines

The Law of Cosines formula for a triangle with sides \(a\), \(b\), \(c\) and angle \(C\) is \(c^{2}=a^{2}+b^{2}-2ab\cos C\). In \(\triangle ABC\), if \(a\) is the side opposite \(\angle A\), \(b = 7\), \(c = 9\) and \(\angle A=117^{\circ}\), then \(a^{2}=b^{2}+c^{2}-2bc\cos A\). Substitute \(b = 7\), \(c = 9\) and \(A = 117^{\circ}\) (\(\cos117^{\circ}\approx - 0.454\)) into the formula: \(a^{2}=7^{2}+9^{2}-2\times7\times9\times\cos117^{\circ}\).

Step2: Calculate the values

First, calculate \(7^{2}=49\), \(9^{2}=81\), and \(2\times7\times9 = 126\). Then \(a^{2}=49 + 81-126\times(-0.454)\). \(a^{2}=49+81 + 57.204\). \(a^{2}=187.204\).

Step3: Find the value of \(a\)

Take the square - root of \(a^{2}\): \(a=\sqrt{187.204}\approx13.68\).

Answer:

B. 13.68