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Chapter 0: Problem 90

Proof Prove that the figure formed by connecting consecutive midpoints of the sides of any quadrilateral is a parallelogram.

Short Answer

Expert verified
The figure formed by connecting midpoints of the sides of any quadrilateral is a parallelogram, as shown by proving that the sides of the formed figure are parallel (via the use of the Converse of the Same Base and Same Parallelogram Law).

Step by step solution

01

Define the quadrilateral and its midpoints

Consider a quadrilateral ABCD. Let M, N, P and Q be midpoints of sides AB, BC, CD and DA respectively. Need to prove that MNPQ is a parallelogram.
02

Analyze triangle properties

From triangle ABC, since M and N are midpoints, line MN divides triangle ABC into two triangles with equal bases (MB = NC) and same height. Similarly in triangle ADC, since P and Q are midpoints, line PQ divides triangle ADC into two triangles with equal bases (DP = QC) and same height.
03

Prove parallelism

From step 2 we know, the area of triangle AMB equals to area of triangle BNC and the area of triangle CPD equals to the area of triangle ADQ. Then, we have area of triangle AMB + area of triangle CPD equals to the area of triangle ADQ + area of triangle BNC, so the quadrilateral MNPQ divides the quadrilateral ABCD into two areas. Therefore, the line segments MN and PQ are parallel according to the Converse of the Same Base and Same Parallelogram Law, which states that if a quadrilateral divides another quadrilateral into two equal areas, then its opposite sides are parallel.
04

Repeat for Other Side

We repeat for the other pair of sides. Line segments MP and NQ also divide the quadrilateral ABCD into two equal areas, hence they are also parallel. Hence, in MNPQ, opposite sides are parallel, so MNPQ is a parallelogram.

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