HOW TO CHECK IF GIVEN FOUR POINTS FORM A RECTANGLE

Example :

Check whether the following four points form a rectangle.

A(-3, -3), B(4, -3), C(4, 2), D(-3, 2)

Solution :

Distance between A and B :

Formula to find the distance between two points :

d = √[(x₂ - x₁)² + (y₂ - y₁)²]

Substitute (x₁, y₁) = A(-3, -3) and (x₂, y₂) = B(4, -3).

= √[(4 + 3)² + (-3 + 3)²]

= √[7² + 0]

= √49

AB = 7 units

Distance between D and C :

= √[(x₂ - x₁)² + (y₂ - y₁)²]

Substitute (x₁, y₁) = D(-3, 2) and (x₂, y₂) = C(4, 2).

= √[(4 + 3)² + (2 - 2)²]

= √[7² + 0]

= √49

DC = 7 units

Distance between A and D :

= √[(x₂ - x₁)² + (y₂ - y₁)²]

Substitute (x₁, y₁) = A(-3, -3) and (x₂, y₂) = D(-3, 2).

= √[(-3 + 3)² + (2 + 3)²]

= √[0 + 5²]

= √25

AD = 5 units

Distance between B and C :

= √[(x₂ - x₁)² + (y₂ - y₁)²]

Substitute (x₁, y₁) = B(4, -3) and (x₂, y₂) = C(4, 2).

= √[(4 - 4)² + (2 + 3)²]

= √[0 + 5²]

= √25

BC = 5 units

From the above workings, AB = DC and AD = BC. 

Opposite sides are equal. 

In the diagram above, consider ΔABC. 

Distance between B and D :

= √[(x₂ - x₁)² + (y₂ - y₁)²]

Substitute (x₁, y₁) = B(4, -3) and (x₂, y₂) = D(-3, 2).

= √[(-3 - 4)² + (2 + 3)²]

= √[49 + 25]

BD = √74

BD² = 74

AB = 7 ----> AB² = 49

AD = 5 ----> AD² = 25

49 + 25 = 74

AB² + AD² = BD²

ΔABC above satisfies Pythagorean Theorem, hence ΔABC is a right triangle with ∠A = 90°. 

Opposite sides are equal and it is proved that one of the vertices has right angle.

So, the given four points form a rectangle. 

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