In trapezoid ABCD with bases AD and BC, the diagonals intersect at point O. AD = 24 cm, BC = 16 cm

univerkov | September 12, 2021 | education

In trapezoid ABCD with bases AD and BC, the diagonals intersect at point O. AD = 24 cm, BC = 16 cm, AC = 12 cm. Find the lengths of the segments OA and OC.

Let us prove that triangle BOS is similar to triangle AOD. Angle BOC = AOD as vertical angles at the intersection of the diagonals AC and BD, angle BCO = DAO as criss-crossing angles at the intersection of parallel straight lines BC and AD secant AC. Then the BOC triangle is similar to the AOD triangle in two corners.

Then BC / OС = AD / AO.

Let OC = X cm, then AO = (12 – X) cm.

16 / X = 24 / (12 – X).

24 * X = 192 – 16 * X.

40 * X = 192.

OS = X = 192/40 = 4.8 cm.

ОА = 12 – 4.8 = 7.2 cm.

Answer: The length of the segment OA = 7.8 cm, OS is 4.8 cm.

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