PROVE THAT THE GIVEN POINTS ARE VERTICES OF RECTANGLE

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

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

Question 1 :

Examine whether the given points 

A(-2, 7), B (5, 4), C (-1, -10) and D (-8, -7)

forms a rectangle.

Solution :

Distance Between Two Points (x1, y1) and (x2 , y2)

√(x2 - x1)2 + (y2 - y1)2

Length of AB :

Here x1 = -2, y1 = 7, x2 = 5  and  y2 = 4

=  √(5-(-2))2+(4-7)2

=  √72+(-3)2

=  √(49+9)

=  √58 units

Length of BC :

Here x1 = 5, y1 = 4, x2 = -1 and  y2 = -10

=  √(-1-5)2 + (-10-4)2

=  √(-6)2 + (-14)2

=  √(36+196)

=  √232 units

Length of CD :

Here x1 = -1, y1 = -10, x2 = -8  and  y2 = -7

=  √(-8-(-1))2 + (-7-(-10))²

=  √(-7)2 + 32

=  √(49+9)

=  √58 units

Length of DA :

Here x1 = -8, y1 = -7, x2 = -2  and  y2 = 7

=  √(-2-(-8))2 + (7-(-7))2

=  √(-2+8)2 + (7+7)2

=  √(36+196)

=  √232 units

Length of AC :

Here x1 = -2, y1 = 7, x2 = -1  and  y2 = -10

=  √(-1-(-2))2 + (-10-7)2

=  √(-1+2)2 + (-17)2

=  √(1+289)

=  √290 units

In triangle ABC,

AC2  =  AB2 + BC2

√2902  =  √582 + √2322

290  =  58 + 232

290  =  290

So, the given points will be the vertices of rectangle.

Question 2 :

Examine whether the given points 

P(-3, 0), Q (1, -2) and R (5, 6) and S (1, 8)

forms a rectangle.

Solution :

Length of PQ :

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

Length of QR :

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

=  √(5-1)2 + (6-(-2))2

=  √42 + (6+2)2

=  √(16 + 64)

=  √80 units

Length of RS :

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

=  √(1-5)2+(8-6)2

=  √(-4)2+22

=  √(16+4)

=  √20 units

Length of SP :

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

=  √(-3-1)2 + (0-8)2

=  √(-4)2+(-8)2

=  √(16+64)

=  √80 units

Length of SP :

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

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

In triangle QRS,

QS²  =  QR² + SR²

(√100)²  =  (√80) ² + (√20)²

100  =  80 + 20

100  =  100

So the given vertices forms a rectangle. 

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