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

| March 15, 2021 | education

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

Let us prove that the triangles BOC and AOD are similar.

Angle BOC = AOD as vertical angles at the intersection of diagonals AC and BD. Angle ОВС = ОDА as criss-crossing angles at the intersection of parallel straight lines АС and ВD of secant ВD.

Then the triangles BOC and AOD are similar in two angles.

Let’s determine the coefficient of similarity of triangles.

K = BC / AD = 16/24 = 2/3.

Let the length of the segment OC = X cm, then OA = 12 – X cm.

OС / OA = 2/3 = K.

X / (12 – X) = 2/3.

24 – 2 * X = 3 * X.

5 * X = 24.

X = OC = 24/5 = 4.8 cm.

ОА = 12 – 4.8 = 7.2 cm.

Answer: OS length = 4.8 cm, OA length = 7.2 cm.



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