Convex quadrilateral ABCD has two pairs of equal between adjacent sides: AB = CD, BC = CD. О is the point

| September 12, 2021 | education

Convex quadrilateral ABCD has two pairs of equal between adjacent sides: AB = CD, BC = CD. О is the point of intersection of the diagonal of the quadrilaterals ABCOD and ABOCD. Compare the P (perimeter) of quads ABOCD and ABOCD.

Let’s write down the perimeters of the pentagons ABCOD and ABOCD.

P1 = AB + BC + OC + OD + AD.

P2 = AB + OB + OC + CD + AD.

By condition, the adjacent sides of the quadrilateral are equal, BC = CD, then:

P1 = (AB + BC + OC + AD) + OD.

P2 = (AB + BC + OC + AD) + OB.

The perimeters of the pentagons differ in the lengths ОВ and ОD.

A quadrangle with equal adjacent sides is a deltoid, or “kite”, in which the diagonals are perpendicular and one of them is divided in half, then OB = OD, and therefore P1 = P2.

Answer: The perimeters are equal.



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