Question

PLEASE HELP WILL AWARD BRAINLIEST TO BEST ANSWER!!!

Answer 1

Answer:

MO = 8 units

MP = 4 units

Step-by-step explanation:

$In\: \triangle MOP, \\\angle P = 90\degree ... (given) \\\angle M = 60\degree ... (given) \\\therefore \angle O = 30\degree(3^{rd} \: \angle \: of\: \triangle) \\Let \huge\purple {MO = x \: units}... (1)\\\therefore MP = \frac{1}{2} \times MO\\(side\: opposite \: to\: 30\degree) \\\\\therefore MP = \frac{1}{2} \times x\\\\\huge\red {\therefore MP = \frac{1}{2} x}.... (2)\\\\\therefore PO = \frac{\sqrt 3}{2} \times MO\\(side\: opposite \: to\: 60\degree) \\\\\therefore PO = \frac{\sqrt 3}{2} \times x\\\\\huge\orange{\therefore PO = \frac{\sqrt 3}{2}x} \\\\\because MP + PO + MO = P(\triangle MOP) \\\\\therefore \frac{1}{2} x+\frac{\sqrt 3}{2}x+ x = 12+4\sqrt 3\\\\\therefore \frac{1}{2} x+x+\frac{\sqrt 3}{2}x= 4(3+\sqrt 3)\\\\\therefore \frac{3}{2} x+\frac{\sqrt 3}{2}x= 4(3+\sqrt 3)\\\\\therefore \frac{(3+\sqrt 3)}{2} x= 4(3+\sqrt 3)\\\\\therefore x = 4(3+\sqrt 3)\times \frac{2}{(3+\sqrt 3)}\\\\\therefore x = 4\cancel{(3+\sqrt 3)}\times \frac{2}{\cancel {(3+\sqrt 3)}}\\\\\therefore x = 4\times 2\\\\\therefore x = 8\\\\\huge\purple {\boxed{\implies MO = 8}} \: \\ [From\: equation \: (1)]\\\\\because MP = \frac{1}{2} x \: \\ [From\: equation \: (2)]\\\\\therefore MP = \frac{1}{2} \times 8\\\\\huge\red {\boxed{\therefore MP = 4 \: units}}$