## VERTICES OF RECTANGLE QUESTION5

Vertices of rectangle question5 :

Here we are going to see practice question on vertices of parallelogram.

Definition of rectangle :

A rectangle is a quadrilateral in which opposite sides are parallel and equal in length. In other words opposite sides of a quadrilateral are equal in length then the quadrilateral is called a rectangle.

## How to test whether the given points are vertices of rectangle ?

(i) First we have to find the length of all sides using distance between two points formula.

(ii) The rectangle can be divided into two right triangles.

(iii) If the given four vertices satisfies those conditions we can say the given vertices forms a rectangle.

## Vertices of rectangle question5 - Solution

Question 5 :

Examine whether the given points  P (-1,1) and Q (0,0) and R (3,3) and S (2,4) forms a rectangle. Solution :

To show that the given points forms a rectangle we need to find the distance between three points.

Distance Between Two Points (x ₁, y₁) and (x₂ , y₂)

√(x₂ - x₁)² + (y₂ - y₁)²

Four points are P (-1,1) and Q (0,0) and R (3,3) and S (2,4)

Distance between the points P and Q

= √(x₂ - x₁)² + (y₂ - y₁)²

Here x₁ = -1, y₁ = 1, x₂ = 0  and  y₂ = 0

=    √(0-(-1))² + (0-1)²

=    √(0+1)² + (-1)²

=    √1² + (-1)²

=    √1 + 1

=    √2 units

Distance between the points Q and R

= √(x₂ - x₁)² + (y₂ - y₁)²

Here x₁ = 0, y₁ = 0, x₂ = 3  and  y₂ = 3

=    √(3-0)² + (3-0)²

=    √(3)² + (3)²

=    √9 + 9

=    √18

=    √18 units

Distance between the points R and S

= √(x₂ - x₁)² + (y₂ - y₁)²

Here x₁ = 3, y₁ = 3, x₂ = 2  and  y₂ = 4

=    √(2-3)² + (4-3)²

=    √(-1)² + (1)²

=    √1 + 1

=    √2 units

Distance between the points S and P

= √(x₂ - x₁)² + (y₂ - y₁)²

Here x₁ = 2, y₁ = 4, x₂ = -1  and  y₂ = 1

=    √(-1-2)² + (1-4)²

=    √(-3)² + (-3)²

=    √9 + 9

=    √18 units

PQ = √2 units

QR = √18 units

RS = √2 units

SP = √18 units

Length of opposite sides are equal.To test whether it forms right triangle we need to find the length of diagonal QS.

Distance between the points Q and S

= √(x₂ - x₁)² + (y₂ - y₁)²

Here x₁ = 0, y₁ = 0, x₂ = 2  and  y₂ = 4

=    √(2-0)² + (4-0)²

=    √(2)² + (4)²

=    √4 + 16

=    √20 units

QS² = QR² + SR²

(√20)² = (√18) ² + (√2)²

20 = 18 + 2

20 = 20

So the given vertices forms a rectangle.

## Try other questions

(1)  Examine whether the given points  A (-3,2) and B (4,2) and C (4,-3) and D (-3,-3) forms a rectangle.

(2)  Examine whether the given points  A (8,3) and B (0,-1) and C (-2,3) and D (6,7) forms a rectangle.

(3)  Examine whether the given points  A (-2,7) and B (5,4) and C (-1,-10) and D (-8,-7) forms a rectangle.

(4)  Examine whether the given points  P (-3,0) and Q (1,-2) and R (5,6) and S (1,8) forms a rectangle.

(5)  Examine whether the given points  P (-1,1) and Q (0,0) and R (3,3) and S (2,4) forms a rectangle.

(6)  Examine whether the given points  P (5,4) and Q (7,4) and R (7,-3) and S (5,-3) forms a rectangle.

(7)  Examine whether the given points  P (0,-1) and Q (-2,3) and R (6,7) and S (8,3) forms a rectangle.

(8)  Examine whether the given points  A (2,-2) and B (8,4) and C (5,7) and D (-1,1) forms a rectangle.

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