Question

a parallelogram has side lengths of 4 and 6 and an angle of measure 55°. what is x, the length of the diagonal, to the nearest whole number? law of cosines: a² = b² + c² - 2bc cos(a)

Explanation:

Step1: Identify values for law of cosines

Let $b = 4$, $c = 6$, and $A=55^{\circ}$. The formula for the law of cosines is $a^{2}=b^{2}+c^{2}-2bc\cos(A)$.

Step2: Substitute values into formula

$a^{2}=4^{2}+6^{2}-2\times4\times6\times\cos(55^{\circ})$
$a^{2}=16 + 36-48\times\cos(55^{\circ})$
Since $\cos(55^{\circ})\approx0.5736$, then $48\times\cos(55^{\circ})\approx48\times0.5736 = 27.5328$.
$a^{2}=16 + 36- 27.5328$
$a^{2}=52-27.5328=24.4672$.

Step3: Find the value of $a$ (which is $x$)

$a=\sqrt{24.4672}\approx4.946$. Rounding to the nearest whole number, $a\approx5$.

Answer:

5