Question

find bc. round to the nearest tenth.
(image of triangle with vertices a, b, c; ab = 6, ac = 13, angle at a is 91°)

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 formula for a triangle with sides \(a\), \(b\), \(c\) and included angle \(C\) is \(c^{2}=a^{2}+b^{2}-2ab\cos(C)\). In triangle \(ABC\), we want to find \(BC\) (let's call it \(a\)), with \(AB = 6\) (let's call it \(c\)), \(AC = 13\) (let's call it \(b\)), and included angle \(A=91^{\circ}\). So the formula becomes \(BC^{2}=AB^{2}+AC^{2}-2\cdot AB\cdot AC\cdot\cos(A)\).

Step2: Substitute the Values

Substitute \(AB = 6\), \(AC = 13\), and \(A = 91^{\circ}\) into the formula:
\(BC^{2}=6^{2}+13^{2}-2\times6\times13\times\cos(91^{\circ})\)
First, calculate \(6^{2}=36\), \(13^{2}=169\), and \(2\times6\times13 = 156\). Then, \(\cos(91^{\circ})\approx - 0.01745\) (using a calculator).
So, \(BC^{2}=36 + 169-156\times(-0.01745)\)
\(BC^{2}=205 + 156\times0.01745\)
\(156\times0.01745\approx2.7222\)
\(BC^{2}\approx205 + 2.7222=207.7222\)

Step3: Take the Square Root

Take the square root of \(207.7222\) to find \(BC\):
\(BC=\sqrt{207.7222}\approx14.4\) (rounded to the nearest tenth)

Answer:

\(14.4\)