Question

In the following figure A, B, C are the midpoints of the sides of $△PQR$. If A, B and Q has the coordinates (3,3), (4,2) and (2,1) respectively, then the coordinates of P,C and R are

• A. (5,4), (4,5) and (4,3)
• B. (6,5), (5,7) and (6,3)
• C. (3,5), (5,6) and (4,3)
• D. (4,5), (5,4) and (6,3)

Answer 1

The correct option is D

(4,5), (5,4) and (6,3)

A and B are midpoints of PQ and RQ respectively.

Hence AB is parallel to PC.

Also, since B and C are midpoints of RQ and PR respectively we have, BC is parallel to PA

$\therefore$ PABC is a parallelogram.

Similarly, ABRC and AQBC are parallelograms.

Now, consider the parallelogram AQBC, as shown below.

Here, we see that : AC = QB

$\angle CAX=\angle BQY$

$\angle ACX=\angle QBY$

$\therefore$ We must have, CX = BY

From the diagram above, we see that, BY = 2 - 1 = 1 unit

$\therefore$ CX = 1 unit

Thus the y-coordinate of C must be 3 + 1 = 4 units.

Also, we have AX = QY

x - coordinate if Y is 4 and that of Q is 2 & hence QY = 2 units.

$\therefore$ AX = 2 units

$\therefore$ x - coordinate of C must be '3 + AX' units, which is, 3 + 2 = 5 units.

$\therefore$ Coordinates of C is(5, 4).

Similarly by considering parallelogram ABRC, we get coordinates of R as (6, 3).

Also, by considering parallelogram PABC, we get coordinates of P as (4, 5).