The bisectors of adjacent angles A and B of parallelogram ABCD intersect sides BC and AD at points M and N

| February 22, 2021 | education

The bisectors of adjacent angles A and B of parallelogram ABCD intersect sides BC and AD at points M and N, respectively. Find the perimeter of the parallelogram if MC = 3m, AN = 8m.

Let us denote the point of intersection of the bisectors by the letter O.

Consider a triangle ABM: the BMO angle is equal to the OAN angle (internal criss-crossing angles with parallel BC and AD and the secant AM), and the OAN angle is equal to the OAB angle (AM is the bisector), which means the BMO angle is equal to the OAB angle. Therefore, the triangle ABM is isosceles, and it means that AB = BM.

ВO will be the median in the triangle ABM, which means that AO = MO.

Consider the ВОМ and NOA triangles:

The ОМВ angle is equal to the OAN angle (internal criss-crossing angles with parallel ВС and АD and secushes АМ). The ВОМ angle is equal to the angle AON (vertical). AO = MO (see above). Therefore, the triangles are equal (in side and two corners).

Hence, ВМ = АN = 8 m. Side BC = ВМ + СМ = 8 + 3 = 11 m.

AB = BM = 8 m.

Let’s calculate the perimeter of the parallelogram:

P (ABCD) = 11 + 8 + 11 + 8 = 38 m.

Answer: the perimeter of the parallelogram is 38 m.



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