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In the diagram, $PQRS$ is a straight line. $R$ is the midpoint of $QS$ and $Q$ is the midpoint of $PS$ . $S$ is $(6, 5)$ and $R$ is $(3, 5)$ . Find the coordinates of $P$ .

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(- 5, 6)

(- 6, 5)

(- 3, 4)

(0, 5)

answer is B.

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Detailed Solution

Given that, R is midpoint of QS and Q is midpoint of PS,

S

(6, 5)

and

R

(3, 5)

According to the midpoint formula, the coordinates of the midpoint of the points

(x_{1}, y_{1})

and

(x_{2}, y_{2})

is given by

(\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2}) .

Let the coordinates of the point

P

and

Q

are

(x, y)

and

(a, b)

respectively.
It is given that

R

is the midpoint of

QS

so, by using midpoint formula,

(\frac{a + 6}{2}, \frac{b + 5}{2}) = (3, 5)

Equating the x an y coordinates,

\Rightarrow \frac{a + 6}{2} = 3

and

\frac{b + 5}{2} = 5

\Rightarrow a + 6 = 6

and

b + 5 = 10

\Rightarrow a = 0

and

b = 5

Hence, coordinate of the point

Q

(0, 5) .

Similarly,

Q

is the midpoint of

PS

so,

(\frac{x + 6}{2}, \frac{y + 5}{2}) = (0, 5)

.
Equating the x and y coordinates,

\Rightarrow \frac{x + 6}{2} = 0

and

\frac{y + 5}{2} = 5

\Rightarrow x + 6 = 0

and

y + 5 = 10

\Rightarrow x = - 6

and

y = 5

Hence, coordinate of the point

P

(- 6, 5) .

Therefore, option (2) is correct.

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