Question

7.b
solve the following problems.
given: △mop,
p_{△mop}=12 + 4\sqrt{3},
m∠p = 90°, m∠m = 60°
find: mp

Explanation:

Step1: Determine side - length ratios

In right - triangle $\triangle MOP$ with $\angle P = 90^{\circ}$ and $\angle M=60^{\circ}$, then $\angle O = 30^{\circ}$. Let $MP = x$. The side opposite the $30^{\circ}$ angle is $MP$, the side opposite the $60^{\circ}$ angle is $OP=\sqrt{3}x$, and the hypotenuse $MO = 2x$.

Step2: Calculate the perimeter

The perimeter $P_{\triangle MOP}=MP + OP+MO$. Substitute the side - length expressions into the perimeter formula: $P_{\triangle MOP}=x+\sqrt{3}x + 2x=(3 + \sqrt{3})x$.

Step3: Solve for $x$

We know that $P_{\triangle MOP}=12 + 4\sqrt{3}$, so $(3+\sqrt{3})x=12 + 4\sqrt{3}$. Factor out a $4$ from the right - hand side: $(3+\sqrt{3})x = 4(3+\sqrt{3})$. Divide both sides by $(3+\sqrt{3})$ to get $x = 4$.

Answer:

$MP = 4$