Question

abby used the law of cosines for △kmn to solve for k. k² = 31² + 53² - 2(31)(53)cos(37°). what additional information did abby know that is not shown in the diagram? law of cosines: a² = b² + c² - 2bccos(a)

Explanation:

Step1: Recall law of cosines formula

The law of cosines is $a^{2}=b^{2}+c^{2}-2bc\cos(A)$, where $a$ is the side opposite angle $A$, and $b$ and $c$ are the other two - sides. In $\triangle KMN$, if $k^{2}=31^{2}+53^{2}-2(31)(53)\cos(37^{\circ})$, then comparing with the law - of - cosines formula, the side opposite the angle used in the cosine term is $k$, and the other two sides are $31$ and $53$. The angle used in the cosine term is the included angle between the sides of lengths $31$ and $53$.

Step2: Identify the angle and sides

The angle between the sides of lengths $31$ and $53$ is $\angle K$, and the side of length $31$ must be $n$ (since $k$ is the side we are solving for). So, $m\angle K = 37^{\circ}$ and $n = 31$.

Answer:

$m\angle K = 37^{\circ}$ and $n = 31$