Question

in $delta def$, $de = 11$, $ef = 9$, and angle $e = 140^{circ}$.
which equation correctly uses the law of cosines to solve for the third side?
$\bigcirc$ $e^{2}=11^{2}+9^{2}-2(11)(9)cos(140^{circ})$
$\bigcirc$ $11^{2}=e^{2}+9^{2}-2e(9)cos(140^{circ})$
$\bigcirc$ $9^{2}=e^{2}+11^{2}-2e(11)cos(140^{circ})$
$\bigcirc$ $e = 11 + 9 - 2(11)(9)cos(140^{circ})$

Explanation:

Step1: Recall Law of Cosines

For $\triangle ABC$ with side $a$ opposite $\angle A$, $b$ opposite $\angle B$, $c$ opposite $\angle C$:
$$c^2 = a^2 + b^2 - 2ab\cos(C)$$

Step2: Match to $\triangle DEF$

Side $e$ is opposite $\angle E$. The sides adjacent to $\angle E$ are $DE=11$ and $EF=9$. Substitute into the formula:
$$e^2 = 11^2 + 9^2 - 2(11)(9)\cos(140^\circ)$$

Step3: Eliminate incorrect options

Other options misassign the side opposite the angle or omit squaring terms.

Answer:

$e^2 = 11^2 + 9^2 - 2(11)(9)\cos(140^\circ)$