SOLUTION: In the diagram to the right, Triangle ABC is isosceles, and Triangle MPQ is equilateral. Find the perimeter, in cm, of Triangle MBP. https://ibb.co/gWnd1MQ

Algebra ->  Length-and-distance -> SOLUTION: In the diagram to the right, Triangle ABC is isosceles, and Triangle MPQ is equilateral. Find the perimeter, in cm, of Triangle MBP. https://ibb.co/gWnd1MQ       Log On


   

Question 1208910: In the diagram to the right, Triangle ABC is isosceles, and Triangle MPQ is equilateral. Find the perimeter, in cm, of Triangle MBP.
https://ibb.co/gWnd1MQ
Answer by ikleyn(52803) About Me  (Show Source):
You can put this solution on YOUR website!
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In the diagram to the right, Triangle ABC is isosceles, and Triangle MPQ is equilateral.
Find the perimeter, in cm, of Triangle MBP.
https://ibb.co/gWnd1MQ
~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

Let x be the side length of the equilateral triangle MPQ.


Consider triangle BMP.


Its angle BPM is  180° - 45° - 60° = 75°.

Its angle B is 45 degrees.


Apply the sine law

    40%2Fsin%28BPM%29 = x%2Fsin%2845%5Eo%29,

or

    40%2Fsin%2875%5Eo%29 = x%2Fsin%2845%5Eo%29.


From it

    x = 40%2A%28sin%2845%5Eo%29%2Fsin%2875%5Eo%29%29.


Next use sin(45^o) = sqrt%282%29%2F2,  sin(75^o) = %28sqrt%286%29+%2B+sqrt%282%29%29%2F4   <<<---=== from your textbook or from the Internet.


Substitute it into the formula for x.  You will get

    x = MP = 40%2A%282%2Asqrt%282%29%29%2F%28sqrt%286%29+%2B+sqrt%282%29%29 = 29.2820323...


To find the side BP, apply the sine law in this form

    40%2Fsin%28BPM%29 = BP%2Fsin%2860%5Eo%29.


It gives

    BP = 40%2A%28sin%2860%5Eo%29%2Fsin%2875%5Eo%29%29 = 40%2A%282%2Asqrt%283%29%29%2F%28sqrt%286%29%2Bsqrt%282%29%29 = 35.86301889... cm



Now the perimeter of the triangle  MBP  is  

    P(MBP) = MP + BP + MB = 29.2820323 + 35.86301889 + 40 = 105.1450512.


It is your  ANSWER,  and you can round it 105.145 cm, approximately.

Solved.