Name: Rectangles and Squares Formed by Four Points
Uploaded: 2019-07-19
Duration: 6 min 3 s
Description: What are the conditions for a quadrilateral to be a rectangle? How does it differ from a square? Unlock your doubts and clear your basics from this video!

Question

In the adjoining figure, $A B C D$ is a square. If $∠ P Q R = 90 °$ and $P B = Q C,$ then find $∠ Q P R$

Accepted Answer

Since, $A B C D$ is a square , then each of its side must be equal.

So, $AB = BC = CD = DA$

Now, $B C = C D$

Subtracting $Q C$ from both sides.

$B C - Q C = C D - Q C \Rightarrow B C - Q C = C D - D R (∵ P B = Q C = D R) \Rightarrow B Q = R C . . . . . . (i)$

In $△ P B Q$ and $△ Q C R$ ,

$P B = Q C$ (Given)

$∠ P Q B = ∠ Q C R = 90 °$

$B Q = Q C$ (From $i$ )

So, $△ P B Q ≅ △ Q C R$ (By SAS)

$\Rightarrow Q P = Q R$ (By CPCT) $. . . . . (i i)$

In $△ P Q R,$

As $Q P = Q R$

$∠ Q R P = ∠ Q P R$ (Angles opposite to equal sides in a triangle are equal)

Also,

$∠ P Q R + ∠ Q R P + ∠ Q P R = 180 ° \Rightarrow 90 ° + ∠ Q P R + ∠ Q P R = 180 ° (∠ Q P R = ∠ Q R P) \Rightarrow 90 ° + 2 ∠ Q P R = 180 ° \Rightarrow 2 ∠ Q P R = 90 ° \Rightarrow ∠ Q P R = 45 °$

Thus, the value of $∠ Q P R$ is $45 °$

M L Aggarwal Solutions for Chapter: Quadrilaterals, Exercise 4: Higher Order Thinking Skills (HOTS)

M L Aggarwal Mathematics Solutions for Exercise - M L Aggarwal Solutions for Chapter: Quadrilaterals, Exercise 4: Higher Order Thinking Skills (HOTS)

Questions from M L Aggarwal Solutions for Chapter: Quadrilaterals, Exercise 4: Higher Order Thinking Skills (HOTS) with Hints & Solutions

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