#### Question

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) ar(PRQ) = 1/2 ar(ARC)

(ii) ar(RQC) = 3/8 ar(ABC)

(iii) ar(PBQ) = ar(ARC)

#### Solution

Take a point S on AC such that S is the mid-point of AC.

Extend PQ to T such that PQ = QT.

Join TC, QS, PS, and AQ.

In ΔABC, P and Q are the mid-points of AB and BC respectively. Hence, by using mid-point theorem, we obtain

PQ || AC and PQ = 1/2AC

⇒ PQ || AS and PQ = AS (As S is the mid-point of AC)

∴ PQSA is a parallelogram. We know that diagonals of a parallelogram bisect it into equal areas of triangles.

∴ ar (ΔPAS) = ar (ΔSQP) = ar (ΔPAQ) = ar (ΔSQA)

Similarly, it can also be proved that quadrilaterals PSCQ, QSCT, and PSQB are also parallelograms and therefore,

ar (ΔPSQ) = ar (ΔCQS) (For parallelogram PSCQ)

ar (ΔQSC) = ar (ΔCTQ) (For parallelogram QSCT)

ar (ΔPSQ) = ar (ΔQBP) (For parallelogram PSQB)

Thus,

ar (ΔPAS) = ar (ΔSQP) = ar (ΔPAQ) = ar (ΔSQA) = ar (ΔQSC) = ar (ΔCTQ) = ar (ΔQBP) ... (1)

Also, ar (ΔABC) = ar (ΔPBQ) + ar (ΔPAS) + ar (ΔPQS) + ar (ΔQSC)

ar (ΔABC) = ar (ΔPBQ) + ar (ΔPBQ) + ar (ΔPBQ) + ar (ΔPBQ)

= 4 ar (ΔPBQ)

⇒ ar (ΔPBQ) = 1/4 ar(ΔABC) ... (2)

(i)Join point P to C.

In ΔPAQ, QR is the median.

`therefore ar(trianglePRQ)=1/2ar(trianglePAQ)=1/2xx1/4ar(triangleABC)=1/8ar(triangleABC)" .........(3)"`

In ΔABC, P and Q are the mid-points of AB and BC respectively. Hence, by using mid-point theorem, we obtain

PQ = 1/2AC

AC = 2PQ ⇒ AC = PT

Also, PQ || AC ⇒ PT || AC

Hence, PACT is a parallelogram.

ar (PACT) = ar (PACQ) + ar (ΔQTC)

= ar (PACQ) + ar (ΔPBQ [Using equation (1)]

∴ ar (PACT) = ar (ΔABC) ... (4)

`ar(triangleARC) = 1/2ar(trianglePAC)" (CR is the median of ΔPAC)"`

`= 1/2xx1/2ar(PACT)" (PC is the diagonal parallelogram PACT)"`

`= 1/4 ar(PACT) = 1/4 ar(triangleABC)`

`rArr 1/2ar(triangleARC) = 1/8ar(triangleABC)`

`rArr1/2ar(triangleARC)=ar(trianglePRQ)" [Using equation (3)] .......(5)"`

(ii)

ar(PACT) = ar(ΔPRQ) + ar(ΔARC) + ar(ΔQTC) + ar(ΔRQC)

Putting the values from equations (1), (2), (3), (4), and (5), we obtain

`ar(ΔABC) = 1/8ar(ΔABC) + 1/4ar(ΔABC) + 1/4ar(ΔABC) + ar(ΔRQC)`

`ar(ΔABC) = 5/8ar(ΔABC) + ar(ΔRQC)`

`ar(ΔRQC) = (1-5/8)ar(ΔABC)`

`ar(ΔRQC) = 3/8ar(ΔABC)`

(iii) In parallelogram PACT,

`ar(ΔARC) = 1/2ar(ΔPAC)" (CR is the median of ΔPAC)"`

`= 1/2xx1/2ar(PACT)" (PC is the diagonal of parallelogram PACT)"`

`= 1/4ar(PACT)`

`= 1/4ar(ABC)`

= ar(ΔPBQ)