O-point of intersection of the parallelogram diagonals ABCD, E and F-midpoints AB and BC, OE = 4cm, ОF = 5cm. Find the perimeter ABCD.
Point O divides the diagonals AC and BD in half.
In triangle ABC, the segment AO = OC and BF = FC. The segment OF connects the midpoints of the sides AC and BC, that is, OF is the middle line of the triangle ABC and therefore is equal to half of the third side of AB:
OF = AB / 2;
AB = 2 * OF = 2 * 5 cm = 10 cm.
In triangle ABD:
AE = EB and OB = OD.
OE is the middle line.
OE = AD / 2;
AD = 2 * OE = 2 * 4 cm = 8 cm.
Perimeter of parallelogram ABCD:
p = 2 * AD + 2 * AB = 2 * 8 cm + 2 * 10 cm = 36 cm.
Answer: 36 cm.
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