Question

In the figure, line pq parallel to side bc then write the ratio in which sides ab and ac are divide proportionately also , give your reason .

Answer 1

Δ's with Same Base and Parallels Possess Equal Area

Step-by-step explanation:

Given:

ΔPQR, in which XY || QR, XY intersects PQ and PR at X and Y respectively.

Construction: Join RX and QY and draw YN perpendicular to PQ and XM perpendicular to PR.

Proof: Since, Area of a triangle = $\frac {1}{2} \times base \times height$

Therefore, ar (ΔPXY) = $\frac {1}{2} \times PX \times YN$ …(1)

Also, ar (ΔPXY) = $\frac {1}{2} \times PY \times XM$ …(2)

Similarly, ar (ΔQXY) = $\frac {1}{2} \times QX \times NY$ …(3)

And, ar (ΔRXY) = $\frac {1}{2} \times YR \times XM$ …(4)

Dividing (1) by (3), we get

$\frac {ar(Triangle\ PXY)}{ar(Triangle\ QXY)} = \frac {\frac {1}{2} \times PX \times YN}{\frac {1}{2} \times QX \times NY} = \frac {PX}{QX}$ ....(5)

Again, dividing (2) by (4), we get

$\frac {ar(Triangle\ PXY)}{ar(Triangle\ RXY)} = \frac {\frac {1}{2} \times PY \times XM}{\frac {1}{2} \times YR \times XM} = \frac {PY}{YR}$....(6)

We know that,

Those triangles having same base as well as same parallel lines are having equal area.

ar(ΔQXY) = ar(ΔRXY) ....(7)

(as and are on same base XY and between same parallel lines XY and QR)

Therefore, from (5), (6) and (7), we get

$\frac {PX}{XQ} = \frac {PY}{YR}$

Hence, proved.