Vertices of Rectangle Worksheet

VERTICES OF RECTANGLE WORKSHEET

Vertices of Rectangle Worksheet :

Here we are going to see some questions on determining whether the given points are the vertices of a rectangle.

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.

Question 1 :

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

Solution :

Solution :

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

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

√(x₂ - x₁)² + (y₂ - y₁)²

Four points are A (-3,2) and B (4,2) and C (4,-3) and D (-3,-3)

Distance between the points A and B

= √(x₂ - x₁)² + (y₂ - y₁)²

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

= √(4-(-3))² + (2-2)²

= √(4+3)² + (0)²

= √7² + 0

= √49

= 7 units

Distance between the points B and C

= √(x₂ - x₁)² + (y₂ - y₁)²

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

= √(4-4)² + (-3-2)²

= √(0)² + (-5)²

= √0 + 25

= √25

= 5 units

Distance between the points C and D

= √(x₂ - x₁)² + (y₂ - y₁)²

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

= √(-3-4)² + (-3-(-3))²

= √(-7)² + (-3+3)²

= √49 + 0

= √49

= 7 units

Distance between the points D and A

= √(x₂ - x₁)² + (y₂ - y₁)²

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

= √(-3-(-3))² + (2-(-3))²

= √(-3+3)² + (2+3)²

= √0 + 5²

= √25

= 5 units

AB = 7 units

BC = 5 units

CD = 7 units

DA = 5 units

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

Distance between the points A and C

= √(x₂ - x₁)² + (y₂ - y₁)²

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

= √(4-(-3))² + (-3-2)²

= √(4+3)² + (-5)²

= √7² + 5²

= √49 + 25

= √74

= √74 units

AC² = AB² + BC²

√74² = 7² + 5²

74 = 49 + 25

74 = 74

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.