**Vertices of rectangle question4 :**

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.

(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 4 :**

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

Distance between the points P and Q =

**√(x₂ - x₁)² + (y₂ - y₁)²**

Here **x₁ = -3, y₁ = 0, x₂ = 1 and y₂ = -2**

**= **
√(1-(-3))² + (-2-0)²

= ** **
√(1+3)² + (-2)²

= ** **
√4² + (-2)²

= √16 + 4

= √20 units

Distance between the points Q and R

= **√(x₂ - x₁)² + (y₂ - y₁)²**

Here **x₁ = 1, y₁ = -2, x₂ = 5 and y₂ = 6**

**= **
√(5-1)² + (6-(-2))²

= ** **
√(4)² + (6+2)²

= ** **
√16 + 8²

= √(16 + 64)

= √80 units

Distance between the points R and S

= **√(x₂ - x₁)² + (y₂ - y₁)²**

Here **x₁ =5, y₁ = 6, x₂ = 1 and y₂ = 8**

**= **
√(1-5)² + (8-6)²

= ** **
√(-4)² + (2)²

= ** **
√16 + 4

= √20 units

Distance between the points S and P

= **√(x₂ - x₁)² + (y₂ - y₁)²**

Here **x₁ = 1, y₁ = 8, x₂ = -3 and y₂ = 0**

**= **
√(-3-1)² + (0-8)²

= ** **
√(-4)² + (-8)²

= ** **
√16 + 64

= √80 units

PQ = √20 units

QR = √80 units

RS = √20 units

SP = √80 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 = 1, y₁ = -2, x₂ = 1 and y₂ = 8**

**= **
√(1-1)² + (8-(-2))²

= ** **
√(0)² + (8+2)²

= ** **
√0 + 10²

= √100

= √100 units

QS² = QR² + SR²

(√100)² = (√80) ² + (√20)²

100 = 80 + 20

100 = 100

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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