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: Find angle O

In a triangle, the sum of interior angles is 180°. So $m\angle O=180^{\circ}-90^{\circ}-60^{\circ}=30^{\circ}$.

Step2: Set up side - length relationships

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

Step3: Calculate the perimeter

The perimeter $P_{\triangle MOP}=MP + OP+MO=x+\sqrt{3}x + 2x=(3 + \sqrt{3})x$.

Step4: Solve for x

Given $P_{\triangle MOP}=12 + 4\sqrt{3}$, we have the equation $(3+\sqrt{3})x=12 + 4\sqrt{3}$. Factor out 4 from the right - hand side: $(3+\sqrt{3})x = 4(3+\sqrt{3})$. Then $x = 4$.

Answer:

$MP = 4$