A straight line is drawn through the point of intersection of the trapezoid diagonals, parallel to the bases

| May 25, 2021 | education

A straight line is drawn through the point of intersection of the trapezoid diagonals, parallel to the bases and intersecting the lateral sides at points E and F, with EF = 8. Find the bases of the trapezoid if their ratio is 4.

The BOC triangle is similar to the AOD triangle by the property of the trapezoid diagonals. Then their similarity coefficient will be equal to: K = AD / BC = 4/1. Then AO / CO = 4. AO = 4 * OC, and the length of AC = AO + OC = 5 * OC.

The ABC triangle is similar to the AOE triangle, since they have a common angle A, OE is parallel to BC, and the AEO angle is equal to the ABC angle.

Then AC / AO = 5 * OC / 4 * OC = 5/4.

Similarly, the triangle BCD is similar to DFO, we get that DB / BO = 5/4.

Then BC / EO = BC / OF, EO = OF.

Then EO = OF = 8/2 = 4 cm.

From similar triangles ABC and AOE BC / EO = 5/4, then BC = 5 * 4/4 = 5 cm.

Then AD = 4 * 5 = 20 cm.

Answer: The lengths of the bases of the trapezoid are 5 cm and 20 cm.



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