Solve the triangle. Find the angles α and β first.
-$\mathrm { a } = 19 , \mathrm {~b} = 16 , \mathrm { c } = 11$
A) $\alpha = 35.3 ^ { \circ } , \beta = 57.3 ^ { \circ } , \gamma = 87.4 ^ { \circ }$
B) $\alpha = 87.4 ^ { \circ } , \beta = 57.3 ^ { \circ } , \gamma = 35.3 ^ { \circ }$
C) $\alpha = 57.3 ^ { \circ } , \beta = 87.4 ^ { \circ } , \gamma = 35.3 ^ { \circ }$
D) $\alpha = 87.4 ^ { \circ } , \beta = 35.3 ^ { \circ } , \gamma = 57.3 ^ { \circ }$

Question

Quizplus · Accepted Answer

The Law of Cosines can be used to find the angles of a triangle when the lengths of all three sides are known. The formula is $\cos(\gamma) = \frac{a^2 + b^2 - c^2}{2ab}$, where $\gamma$ is the angle opposite side $c$, and $a$, $b$, and $c$ are the lengths of the sides. Applying this formula and similar ones for angles $\alpha$ and $\beta$, we can calculate the angles based on the given side lengths. The correct angles correspond to the side lengths in a way that the largest angle is opposite the longest side, and the smallest angle is opposite the shortest side. Given $a = 19$, $b = 16$, and $c = 11$, the largest side $a$ suggests that $\alpha$ will be the largest angle, which matches option B.

Solve the Triangle a=19, b=16,c=11\mathrm { a } = 19 , \mathrm {~b} = 16 , \mathrm { c } = 11a=19, b=16,c=11

Solve the Triangle $\mathrm { a } = 19 , \mathrm {~b} = 16 , \mathrm { c } = 11$