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) $94.61 ^ { \circ }$
C) $95.95$
D) $93.42$
E) $97.13 ^ { \circ }$

Question

Quizplus · Accepted Answer

To find the measure of the angle at vertex $B$, we use the dot product formula for vectors $\vec{AB} \cdot \vec{BC} = |\vec{AB}||\vec{BC}|\cos(\theta)$, where $\theta$ is the angle at $B$. First, find vectors $\vec{AB}$ and $\vec{BC}$ by subtracting coordinates: $\vec{AB} = (-1-6, 3-1) = (-7, 2)$ and $\vec{BC} = (-4-(-1), -5-3) = (-3, -8)$. The dot product $\vec{AB} \cdot \vec{BC} = (-7)(-3) + (2)(-8) = 21 - 16 = 5$. The magnitudes are $|\vec{AB}| = \sqrt{(-7)^2 + 2^2} = \sqrt{49 + 4} = \sqrt{53}$ and $|\vec{BC}| = \sqrt{(-3)^2 + (-8)^2} = \sqrt{9 + 64} = \sqrt{73}$. Thus, $\cos(\theta) = \frac{5}{\sqrt{53}\sqrt{73}}$. Solving for $\theta$ gives $\theta = \cos^{-1}\left(\frac{5}{\sqrt{53}\sqrt{73}}\right)$, which calculates to approximately $94.61^\circ$.

Use Vectors to Find the Measure of the Angle at Vertex