# Q : 7     P and  Q are respectively the mid-points of sides AB and BC of a triangle ABC and R is the mid-point of AP, show that             (i)

We have ABC such that P, Q and R are the midpoints of the side AB, BC and AP respectively. Join PQ, QR, AQ, PC and RC as shown in the figure.
Now, in  APC,
Since R is the midpoint. So, RC is the median of the APC
Therefore,  ar(ARC) = 1/2 . ar (APC)............(i)

Also, in ABC, P is the midpoint. Thus CP is the median.
Therefore, ar(APC) = 1/2. ar (ABC)............(ii)

Also, AQ is the median of ABC
Therefore, 1/2. ar (ABC) = ar (ABQ)............(iii)

In APQ, RQ is the median.
Therefore, ar (PRQ) = 1/2 .ar (APQ).............(iv)

In ABQ, PQ is the median
Therefore, ar(APQ) = 1/2. ar(ABQ).........(v)

From eq (i),
...........(vi)
Now, put the value of ar(APC) from eq (ii), we get
Taking RHS;

(from equation (iii))

(from equation (v))

(from equation (iv))

Hence proved.

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