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. {i} Explain why the y coordinate of D is 6. {ii} given that the x coordinate of D is h, express the gradients of AD and CD interms of h. {iii} calculate the x coordinates of D and B. {iv} calculate the area of the rectangle ABCD. End of Quiz 2 (
Answers
Given : 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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Step-by-step explanation:
Let us assume that 5 -√3 is rational.
Let ,
5 - √3 = r , where "r" is rational
5 - r = √3
Here,
LHS is purely rational.
But on the other hand ,RHS is irrational.
This leads to a contradiction.
Hence,5-√3 is irrational