Use vectors to find the measure of the angle at vertex $B$ of triangle $A B C$, when $A = ( 6,1 ) , B = ( - 1,3 )$, and $C = ( - 4 , - 5 )$. Round answer to two decimal places.
A) 92.52
B) 94.61°
C) 95.95
D) 93.42
E) 97.13°

Question

Quizplus · Accepted Answer

To find the angle at vertex $B$, we first find the vectors $\overrightarrow{BA}$ and $\overrightarrow{BC}$. $\overrightarrow{BA} = A - B = (6 - (-1), 1 - 3) = (7, -2)$ and $\overrightarrow{BC} = C - B = (-4 - (-1), -5 - 3) = (-3, -8)$. The angle $\theta$ between two vectors can be found using the dot product formula: $\overrightarrow{BA} \cdot \overrightarrow{BC} = |\overrightarrow{BA}| |\overrightarrow{BC}| \cos(\theta)$, where $\overrightarrow{BA} \cdot \overrightarrow{BC} = (7)(-3) + (-2)(-8) = -21 + 16 = -5$, $|\overrightarrow{BA}| = \sqrt{7^2 + (-2)^2} = \sqrt{49 + 4} = \sqrt{53}$, and $|\overrightarrow{BC}| = \sqrt{(-3)^2 + (-8)^2} = \sqrt{9 + 64} = \sqrt{73}$. Thus, $\cos(\theta) = \frac{-5}{\sqrt{53} \sqrt{73}}$, and $\theta = \cos^{-1}\left(\frac{-5}{\sqrt{53} \sqrt{73}}\right)$, which approximately equals $94.61^\circ$.

Use Vectors to Find the Measure of the Angle at Vertex