Question

Points A (3,9), B = (1, 1) C = (5, 3) and D =(a ,b) lie in the first quadrant and are the vertices of
quadrilateral ABCD. The quadrilateral formed by joining the mid points of AB, BC, CD and DA is a
square. The sum of the coordinates of point D?
(a) 8
(b) 7
( 10
(d) 12​

Answer 1

the sum of the coordinates of point D is 10

Answer 2

The sum of the coordinate of point D is 10

C is correct

Step-by-step explanation:

Please find the attachment for figure.

E is mid point of AB

$E=\left(\dfrac{3+1}{2},\dfrac{9+1}{2}\right)$

$E=\left(2,5\right)$

F is mid point of BC

$F=\left(\dfrac{5+1}{2},\dfrac{3+1}{2}\right)$

$F=\left(3,2\right)$

Slope of EF

$\text{Slope of EF}=\dfrac{5-2}{2-3}=-3$

G is mid point of CD

$G=\left(\dfrac{a+5}{2},\dfrac{b+3}{2}\right)$

Slope of FG

$\text{Slope of FG}=\dfrac{\frac{b+3}{2}-2}{\frac{a+5}{2}-3}$

$\text{Slope of FG}=\dfrac{b-1}{a-1}$

EF and FG is side of square. Side of square is perpendicular to each other.

If two lines are perpendicular then product of their slopes is -1

$\dfrac{b-1}{a-1}\times -3=-1$

$3b-3=a-1$

$a=3b-2$

The equation of line FG and Equation of line CD.

The intersection of line EG and line CD is G

So, G(6,3)

$(6,3)=\left(\dfrac{a+5}{2},\dfrac{b+3}{2}\right)$

$a=7,b=3$

Sum of a+b = 7 + 3

a + b = 10

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