Name: Introduction to Quadrilaterals
Uploaded: 2019-07-19
Duration: 5 min 30 s
Description: We all watch TV and play games on the computer. Have you ever observed the shape of your TV or computer monitor? They are both quadrilaterals. Let’s learn ab...

Question

Prove that the bisectors of interior angles of a rectangle form a square.

Accepted Answer

Given: $A B C D$ is a rectangle.

To Prove: $P Q R S$ is a square.

Proof: Let $P Q R S$ be the quadrilateral formed by bisectors of $∠ A, ∠ B, ∠ C$ and $∠ D$ .

Since $D S$ bisects $∠ D$ and $AS$ bisects $∠ A$ , therefore,

$∠ D A S + ∠ A D S = \frac{1}{2} ∠ A + \frac{1}{2} ∠ D$

$= \frac{1}{2} (∠ A + ∠ D) = \frac{1}{2} \times (90 ° + 90 °) = 90 °$

Also, $∠ D A S + ∠ A D S + ∠ D S A = 180 °$

(Angle sum property of a triangle)
$\Rightarrow 90 ° + ∠ D S A = 180 °$

$\Rightarrow ∠ D S A = 90 °$

$∴ ∠ P S R = 90 °$ (Vertically opposite angles are equal)

Similarly, it can be shown that

$∠ P Q R = 90 °$ , $∠ S R Q$ $= 90 °$ and $∠ S P Q = 90 °$

So, $P Q R S$ is a quadrilateral in which all angles are right angles.

$\Rightarrow ∠ P = ∠ Q = ∠ R = ∠ S = 90 °$

Hence, $P Q R S$ is definitely a rectangle.

Now, $∠ A D S = ∠ S A D = 45 °$

So, $∆ A D S$ is an isosceles triangle.

Hence, a line passing through midpoint $E$ of the side $A D$ and parallel to $D C$ will bisect the angle $∠ A S D and ∠ P S R$

Similarly, a line passing through midpoint $F$ of the side $B C$ and parallel to $D C$ will bisect the angle $∠ B Q C and ∠ R Q P$ .

Clearly, the diagonal $S Q$ is bisecting the angles of the rectangle $P Q R S .$

Hence, $P Q R S$ is a square.

Prove that the bisectors of interior angles of a rectangle form a square.

Important Questions on Rectilinear Figures [Quadrilaterals : Parallelogram, Rectangle, Rhombus, Square and Trapezium]

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