Name: Example on Thales Theorem
Uploaded: 2019-07-19
Duration: 4 min 26 s
Description: If the midpoint of two sides of a triangle is joined, then this line segment is parallel to the third side. Let's explore more by an example in this video.

Question

Prove that the quadrilateral formed by joining the midpoints of a quadrilateral is a parallelogram. What if the original quadrilateral is a rectangle? What if it is a rhombus?

Accepted Answer

To prove: The quadrilateral formed by joining the midpoints of a quadrilateral is a parallelogram.

Proof:

Consider the below diagram:

$P, Q, R and S$ are the midpoints of the sides of the quadrilateral $A B C D$ . Draw the diagonals $A C, B D$ .

The length of the line joining the midpoints of two sides of a triangle is half the length of the third side.

In $Δ A B C, P Q = \frac{1}{2} A C$ and in $Δ A D C, S R = \frac{1}{2} A C$

$∴ P Q = S R$

In $Δ A B D, P S = \frac{1}{2} B D$ and in $Δ B C D, Q R = \frac{1}{2} B D$

$∴ P S = Q R$

In quadrilateral $P Q R S$ , both pairs of opposite sides are equal. So it is a parallelogram.

Hence, proved.

Case I: The original quadrilateral is a rectangle. (Given)

Then $P Q = \frac{1}{2} A C, R S = \frac{1}{2} A C$

$∴ R S = P Q = \frac{1}{2} A C$

Also $P S = \frac{1}{2} B D, Q R = \frac{1}{2} B D$

$∴ P S = Q R = \frac{1}{2} B D = \frac{1}{2} A C$

$∴ P Q = R S = P S = Q R$

All sides of the quadrilateral $P Q R S$ are equal. So it is a rhombus.

Case II: The original quadrilateral is a rhombus. (Given)

Since $A B C D$ is a rhombus. $A C$ perpendicular to $B D$ .

$P Q and S P$ are perpendicular, i.e., $∠ S P Q = 90 °$

One angle of the parallelogram $P Q R S$ is $90 °$ and therefore it is a rectangle.

Prove that the quadrilateral formed by joining the midpoints of a quadrilateral is a parallelogram. What if the original quadrilateral is a rectangle? What if it is a rhombus?

Important Questions on Parallel Lines

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