Prove that the trapezoids into which the segment connecting the midpoints of its bases

univerkov | July 14, 2021 | education

Prove that the trapezoids into which the segment connecting the midpoints of its bases splits the given trapezoid have equal areas.

Since, by condition, points K and M are the middle of the bases BC and AD, then BK = CK, AM = DM.

Since BC is parallel to AD as the base of the trapezoid, the KM segment divides the ABCD trapezoid into two trapezoids, ABKM and CDMK.

Let us construct the height of KH, which is the height of all three trapezoids.

Then the area of the trapezoid ABKM is equal to: Sawkm = (BK + AM) * KН / 2.

The area of the trapezoid СDМК is equal to: Sсдмк = (СK + DМ) * КН / 2.

(BK + AM) = (CK + DM).

Then Savkm = Ssdmk, as required.

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