Q. 64 - Calculus

Question

Let Quad(PQRS) denote the quadrilateral in the XY-plane with vertices P, Q, R, and S. If P' is the midpoint of side PQ, Q' is the midpoint of side QR, R' is the midpoint of side RS, and S' is the midpoint of side SP, prove that Quad(P' Q' R' S' ) is a parallelogram. 

Step-by-Step Solution

Answer

It has been proven that Quad(P' Q' R' S' ) is a parallelogram. 

Step 1: Draw the diagram

According to the question, the diagram is shown below,

In the above diagram,  P' is the midpoint of side PQ, Q' is the midpoint of side QR, R' is the midpoint of side RS, and S' is the midpoint of side SP.

Step 2: Proof

We will use the mid-point theorem here. It states that the line segment joining the mid-points of any two sides of the triangle is parallel to the third side and is half of it. 

 In ΔPSR, S' and R' are the mid-points of sides PS and SR respectively. Thus, by using the mid-point theorem

∴ S'R' || PR and S'R'= 1/2PR ... (1)

In ΔPQR, P' and Q' are mid-points of sides PQ and QR. Therefore, by using the mid-point theorem,

P'Q' || PR and P'Q' = 1/2 PR ... (2)

Using Equations (1) and (2), we obtain P'Q' || S'R' and P'Q' = S'R' ... (3)

∴ P'Q' = S'R'

From Equation (3), we obtained P'Q' || S'R' and P'Q' = S'R'

Clearly, one pair of opposite sides of quadrilateral P'Q'R'S' is parallel and equal. Hence, P'Q'R'S' is a parallelogram.