Use vectors to find the measure of the angle at vertex B of triangle ABC,when $A = ( 5,1 )$ , $B = ( - 5,1 )$ ,and $C = ( - 3 , - 5 )$ .Round answer to two decimal places.
A)69.48°
B)71.57°
C)72.91°
D)70.38°
E)74.09°

Question

Quizplus · Accepted Answer

We can use the dot product of vectors AB and BC to find the angle at vertex B:
$\cos\angle B = \frac{\overrightarrow{AB} \cdot \overrightarrow{BC}}{\left\lVert \overrightarrow{AB} ightVert \cdot \left\lVert \overrightarrow{BC} ightVert}$
$\overrightarrow{AB} = \begin{pmatrix} 3 \ 2 \end{pmatrix} - \begin{pmatrix} 1 \ 1 \end{pmatrix} = \begin{pmatrix} 2 \ 1 \end{pmatrix}$
$\overrightarrow{BC} = \begin{pmatrix} 1 \ 5 \end{pmatrix} - \begin{pmatrix} 3 \ 2 \end{pmatrix} = \begin{pmatrix} -2 \ 3 \end{pmatrix}$
$\left\lVert \overrightarrow{AB} ightVert = \sqrt{2^2 + 1^2} = \sqrt{5}$
$\left\lVert \overrightarrow{BC} ightVert = \sqrt{(-2)^2 + 3^2} = \sqrt{13}$
$\overrightarrow{AB} \cdot \overrightarrow{BC} = (2)(-2) + (1)(3) = -1$
$\cos\angle B = \frac{-1}{\sqrt{5}\sqrt{13}} = \frac{-1}{\sqrt{65}}$
$\angle B = \cos^{-1}\left( \frac{-1}{\sqrt{65}} ight) \approx 71.57$ degrees
Rounding to two decimal places gives the answer of choice B, 71.57 degrees.

Use Vectors to Find the Measure of the Angle at Vertex