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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)

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B

(6,5), (5,7) and (6,3)

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C

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

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D

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

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Solution

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

PABC is a parallelogram.

Similarly, ABRC and AQBC are parallelograms.

Now, consider the parallelogram AQBC, as shown below.

Here, we see that : AC = QB

CAX=BQY

ACX=QBY

We must have, CX = BY

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

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.

AX = 2 units

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

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).


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