Question

Given: △MOP, Perimeter of △MOP=12+4 sqr rt. 3 m∠P=90°, m∠M=60° Find: MP

Answer 1

mp=4 cuz in a 30-60-90 degree right triangle if the shorter leg is x, then the longer leg (po) is xsqrt3. Because the perimeter is 12+4sqrt3, that means the longer leg is probably 4sqrt3. because of this, then the shorter leg, mp is 4. Since the hypotenuse is equal to 2x, then it would be 2*4=8=PO.

Answer 2

Consider triangle MOP. This triangle is right triangle, because m∠P=90°.

Since m∠P=90°, m∠M=60°, then m∠O=180°-90°-60°=30°.

The leg opposite to the angle with measure 30° is half of the hypotenuse. Let the leg MP=x, then the hypotenuse MO=2x. By the Pythagorean theorem,

`PO^2+MP^2=MO^2\Rightarrow PO^2=(2x)^2-x^2=4x^2-x^2=3x^2\Rightarrow PO=√(3)x.`

The perimeter of triangle MOP is

`P_(MOP)=MO+PO+MP=x+2x+√(3)x=3x+√(3)x.`

Since
`P_(MOP)=12+4√(3),` then
`x=4.`

Answer: MP=4 units.