The bisector of the parallelogram ABCD, drawn from the vertex A, bisects the opposite side

| January 14, 2021 | education

The bisector of the parallelogram ABCD, drawn from the vertex A, bisects the opposite side. BC-22 centimeters. Find the perimeter of the parallelogram.

Opposite angle A there are two sides – BC and CD, the bisector can intersect both BC and AD. Therefore, the problem has two solutions.
1) Let the bisector of angle A intersect side BC at point E (BE = CE).
Consider a triangle ABE: the angle BEA is equal to the angle EAD (internal cross-lying angles with parallel BC and AD and secant AE), and the angle EAD is equal to the angle EAB (AE is the bisector). Therefore, the angle BEA is equal to the angle EAB, which means that the triangle ABE is isosceles (in an isosceles triangle, the angles at the base are equal). Hence, BE = AB.
BE = 1/2 BC = 11 cm.
Hence RAВСD = (22 + 11) * 2 = 66 cm.
2) Let the bisector of angle A intersect СD at point E (CE = DE).
Consider the triangle AED: the angle AED is equal to the angle EAB (internal criss-crossing angles with parallel AB and СD and secant AE), and the angle EAD is equal to the angle EAB (AE is the bisector).
Therefore, the angle EAD = the angle DEA, which means that the triangle AED is isosceles, and hence: AD = DE. Since AD = BC = 22, it means DE = 22, and the side of the СD = 22 * ​​2 = 44 cm.
Hence RAВСD = (22 + 44) * 2 = 132 cm.



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