**Vertices of rectangle worksheet :**

Here we are going to see some question to verify how to examine whether the given points are the vertices of 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.

(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.

(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.

- Solution for vertices of rectangle question1
- Solution for vertices of rectangle question2
- Solution for vertices of rectangle question3
- Solution for vertices of rectangle question4
- Solution for vertices of rectangle question5
- Solution for vertices of rectangle question6
- Solution for vertices of rectangle question7
- Solution for vertices of rectangle question8

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