**How to Check if Given Four Points Form a Rectangle ?**

Here we are going to see, how to check if given four points form a rectangle.

In order to prove the given points form a rectangle, we have to follow the steps given below.

(i) Plot the given points in the graph and join the points.

(ii) Find the length of all sides.

(iii) In a rectangle, the length of opposite sides will be equal.

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

(v) If the given four vertices satisfies the conditions (iii) and (iv), we may decide that the given points form 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 :**

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 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 B and C :**

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 C and D :**

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 D and A :**

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

Since the diagonal AC divides the rectangle into two right triangle, the given vertices will form 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. Solution

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

(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

(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

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

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

(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

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