Question

Parallelogram Show that if the midpoints of consecutive sides of any quadrilateral are connected, the result is a parallelogram.

Short answer

By using the midpoint theorem on the triangles made by the quadrilateral and connecting the midpoints, we find that the lines connecting these midpoints are equal and parallel, establishing that the figure formed is a parallelogram.

Step-by-step solution

Identify and Connect Midpoints

Firstly, identify and connect the midpoints of consecutive sides of the quadrilateral. Let's label the midpoints of the quadrilateral ABCD as M, N, P, and Q in clockwise direction.

Apply Mid-point Theorem

According to the midpoint theorem in geometry, the line-segment joining the midpoints of any two sides of a triangle is parallel to the third side and is half of it. Therefore, MP is parallel and equal to half of AC.

Repeat Mid-point Theorem Application

Now apply the mid-point theorem again for triangles ABC and ADC. As a result, we find that NQ (connecting the midpoints of sides BC and CD) is also parallel and equal to half of AC.

Infer Parallelogram

To designate a quadrilateral as a parallelogram, its opposite sides should be parallel. Here, since MP and NQ are both parallel and equal to each other, quadrilateral MPNQ is a parallelogram.