SOLUTION: ABC is a right angle isosceles triangle, angle BCA = 90, with BC as the base and AB as hypotenuse side AC = BC. Point M is at midpoint on hypotenuse such that BM=MA=36cm. P and Q a

Algebra ->  Trigonometry-basics -> SOLUTION: ABC is a right angle isosceles triangle, angle BCA = 90, with BC as the base and AB as hypotenuse side AC = BC. Point M is at midpoint on hypotenuse such that BM=MA=36cm. P and Q a      Log On


   

Question 1148503: ABC is a right angle isosceles triangle, angle BCA = 90, with BC as the base and AB as hypotenuse side AC = BC. Point M is at midpoint on hypotenuse such that BM=MA=36cm. P and Q are points on sides BC and AC respectively. An equilateral triangle is formed by joining MPQ. Find the area of equilateral triangle MPQ.
Answer by ikleyn(52786) About Me  (Show Source):
You can put this solution on YOUR website!
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1.  Make a sketch.



2.  Due to symmetry,  angle BMP = angle AMQ = 60°.



3.  Consider triangle BMP.

    Its side BM is 36 cm;  its side MP is unknown; let it be "a" cm long.

    Its angle BMP is 60°; its angle MBP is 45°.

    Its angle MPB = 180° - 60° - 45° = 105°.



4.  Apply the sine law theorem to triangle BMP.


        36%2Fsin%28105%5Eo%29%29 = a%2Fsin%2845%5Eo%29.    (1)


    Use  sin(105°) = sin(180°-105°) = sin(75°) = sin(45°+30°) = sin(45°)*cos*30°) + cos(45°)*sin(30°) = %28sqrt%282%29%2F2%29%2A%28sqrt%283%29%2F2%29+%2B+%28sqrt%282%29%2F2%29%2A%281%2F2%29 = %28sqrt%286%29%2Bsqrt%282%29%29%2F4.


    Use  sin(45°) = sqrt%282%29%2F2.


    Then from (1)  you get


        a = %2836%2Asin%2845%5Eo%29%29%2Fsin%28105%5Eo%29 = 36%2A%28%28sqrt%282%29%2F2%29%2F%28%28sqrt%286%29%2Bsqrt%282%29%29%2F4%29%29 = 36%2A%28%282%2Asqrt%282%29%29%2F%28sqrt%286%29%2Bsqrt%282%29%29%29.   (2)


    It is the final formula for the unknown side length "a".


    If in addition to the final formula (2) you need its numerical value,

    here it is   a = 26.354 centimeters,  with 3 correct decimal places after the decimal dot.


     Now, if you want to find the area of the triangle MPQ, use the formula


        area = %28a%5E2%2Asqrt%283%29%29%2F4 = %2826.354%5E2%2Asqrt%283%29%29%2F4 = 300.74 cm^2.

Solved.