Name: The Area Around the Edges
Uploaded: 2019-07-19
Duration: 4 min 16 s
Description: The area enclosed between two figures drawn one inside the other is the area around the edges. Learn how to calculate the area around the edges of a given sh...

Question

A rectangle is formed by joining the midpoints of the sides of a rhombus. Show that the area of rectangle is half the area of the rhombus.

Accepted Answer

Given: $P Q R S$ is a rhombus and $A, B, C$ and $D$ are the midpoint of the sides $P Q, Q R, R S$ and $P S$ respectively.
To Prove: Area of the rectangle is half of the area of the rhombus.

$a r (A B C D) = \frac{1}{2} a r (P Q R S)$

Proof:

In the triangle $P Q R$ , $A$ is the midpoint of $P Q$ and $B$ is the midpoint of $Q R$ .
$∴ A B ∥ P R$
$A B = \frac{1}{2} \times P R$ [by the midpoint theorem]

$\Rightarrow P R = 2 A B \to (1)$
And in the triangle $P Q S$ , $A$ is the midpoint of $P Q$ and $D$ is the midpoint of $P S$ .
$∴ A D ∥ Q S$
$A D = \frac{1}{2} \times Q S$ ….[by the midpoint theorem]

$\Rightarrow Q S = 2 A D \to (2)$

Area of the rhombus $= \frac{1}{2} \times$ product of the diagonals.
$= \frac{1}{2} \times P R \times Q S$
$= \frac{1}{2} \times 2 A B \times 2 A D$ [by equation (1) and (2)]
$= 2 \times A B \times A D$
$= 2 \times$ area of the rectangle
So, area of rectangle $A B C D = \frac{1}{2}$ of area of rhombus $P Q R S$
Thus, area of the rectangle is half of the area of the rhombus.

A rectangle is formed by joining the midpoints of the sides of a rhombus. Show that the area of rectangle is half the area of the rhombus.

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