Answer
The diagonal opposite the angle of $58^{\circ}$ has a length of 5.2 cm
The diagonal opposite the angle of $122^{\circ}$ has a length of 8.8 cm
Work Step by Step
Let $a = 4.0~cm$, let $b = 6.0~cm$, and let angle $C_1 = 58^{\circ}$.
We can use the law of cosines to find $c_1$, the diagonal opposite the angle $C_1$:
$c_1^2 = a^2+b^2-2ab~cos~C_1$
$c_1 = \sqrt{a^2+b^2-2ab~cos~C_1}$
$c_1 = \sqrt{(4.0~cm)^2+(6.0~cm)^2-(2)(4.0~cm)(6.0~cm)~cos~58^{\circ}}$
$c_1 = \sqrt{26.56~cm^2}$
$c_1 = 5.2~cm$
Let $a = 4.0~cm$, let $b = 6.0~cm$, and let angle $C_2 = 122^{\circ}$.
We can use the law of cosines to find $c_2$, the diagonal opposite the angle $C_2$:
$c_2^2 = a^2+b^2-2ab~cos~C_2$
$c_2 = \sqrt{a^2+b^2-2ab~cos~C_2}$
$c_2 = \sqrt{(4.0~cm)^2+(6.0~cm)^2-(2)(4.0~cm)(6.0~cm)~cos~122^{\circ}}$
$c_2 = \sqrt{77.44~cm^2}$
$c_2 = 8.8~cm$
