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Q.

In the Cartesian plane above, the points P, Q and S are three vertices of the rectangle PQRS. Find the coordinates of R.

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a

(5,6)

b

(8,2)

c

(4,6)

d

(6,4)

answer is A.

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Detailed Solution

We are given the graph having three coordinates of a rectangle PQRS.
We first find the coordinates of points P, Q and S.
Point P:
Distance of point P from origin on x-axis is 3
Distance of point P from origin on y-axis is 0
Point becomes P(3,0)

Point Q:
Distance of point Q from origin on x-axis is 1
Distance of point Q from origin on y-axis is 2
Point becomes Q(1,2)

Point S:
Distance of point S from origin on x-axis is 7
Distance of point S from origin on y-axis is 4
Point becomes S(7,4)
seo

seo

Let the midpoint of the diagonals PR and QS be O
Use the formula of the midpoint of two points to write the value of O from both diagonals. Midpoint of two points (x,y) and (a,b) is given by

(\frac{x + a}{2}, \frac{y + b}{2})

Then the value of mid-point from diagonals PR and QS will be the same, as the point O is the mid-point of both diagonals.
⇒Midpoint of PR = Midpoint of QS

\Rightarrow (\frac{x + 3}{2}, \frac{0 + y}{2}) = (\frac{7 + 1}{2}, \frac{4 + 2}{2})

\Rightarrow (\frac{x + 3}{2}, \frac{0 + y}{2}) = (\frac{8}{2}, \frac{6}{2})

\Rightarrow (\frac{x + 3}{2}, \frac{0 + y}{2}) = (4, 3)

Equate the x-coordinate in LHS to x-coordinate in RHS and y-coordinate in LHS to y-coordinate in RHS.

\Rightarrow \frac{x + 3}{2} = 4 and \frac{y}{2} = 3

\Rightarrow x + 3 = 8 and  y = 6

⇒x=8−3 and y=6
⇒x=5 and y=6
So, the point R becomes (5,6)
∴ Coordinates of point R are (5,6)
∴ Option 1 is correct.

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