Question

find bc? round to the nearest tenth.

Explanation:

Step1: Identify the Law to Use

We have a triangle with two sides and the included angle. So we use the Law of Cosines. The Law of Cosines states that for a triangle with sides \(a\), \(b\), \(c\) and the included angle \(C\) between sides \(a\) and \(b\), \(c^{2}=a^{2}+b^{2}-2ab\cos(C)\). In triangle \(ABC\), we know \(AB = 13\), \(AC=7\), and \(\angle A = 140^{\circ}\). We want to find \(BC\), let's call \(BC = x\). So by the Law of Cosines: \(x^{2}=13^{2}+7^{2}-2\times13\times7\times\cos(140^{\circ})\)

Step2: Calculate the Values

First, calculate \(13^{2}=169\), \(7^{2} = 49\). Then calculate \(2\times13\times7=182\). Now, \(\cos(140^{\circ})=\cos(180^{\circ} - 40^{\circ})=-\cos(40^{\circ})\approx - 0.7660\)

So substitute these values into the formula:

\(x^{2}=169 + 49-182\times(- 0.7660)\)

\(x^{2}=218+182\times0.7660\)

Calculate \(182\times0.7660\approx182\times0.766 = 139.412\)

Then \(x^{2}=218 + 139.412=357.412\)

Step3: Find the Square Root

Take the square root of \(357.412\) to find \(x\): \(x=\sqrt{357.412}\approx18.9\) (rounded to the nearest tenth)

Answer:

\(18.9\)