The diagram shows a rectangle ABCD. The point A is (0,-2) and C is (12,14). The diagonal BD is parallel to the x-axis. Explain why the y coordinate of D is 6. The x coordinate of D is h. Given that the x coordinate of D is h, express gradients of AD and CD in terms of h. Calculate the x coordinates of D and B. Calculate the area of rectangle ABCD.

The point A is (0,−2) and C is (12,14). The diagonal BD is parallel to the x-axis.

To find : Explain why the y coordinate of D is 6

Solution:

The point A is (0,−2) and C is (12,14).

Diagonals of rectangle bisect each other

Hence Diagonal BD will pa ss through mid point of A & C

Mid point of A & C  

= (0 + 12)/2 , ( - 2 + 14)/2

= 6 , 6

diagonal BD is parallel to the x-axis.

Hence slope = 0

Equation of BD

=> y - 6 = 0(x - 6)

=> y = 6

Hence  the y coordinate of D is 6

x coordinate of D is h,

=> D = ( h , 6)

A  (0,−2)

gradients of AD   = 8/h

C   (12,14).

gradients of CD   = 8/(12-h)

gradients of AD * gradients of CD = - 1

=> (8/h)  ( 8/(12 - h)  = -1

=> h² - 12h  - 64 = 0

=> (h - 16)(h + 4) = 0

=> h = 16 , h = - 4

Hence B = ( 16 , 6)

D = (-4 , 6)

x coordinate of B & D are  - 4 & 16

A  (0,−2) B = ( 16 , 6)  => AB = 8√5

A  (0,−2) D = ( -4 , 6)  => AC = 4√5

Area of rectangle ABCD = 8√5 x  4√5   = 160 sq units

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