In the given figure, PQ || BA and PR || CA. If PD=12 cm then find BD×CD.

A
${144 cm}^{2}$
B
${135 cm}^{2}$
C
${124 cm}^{2}$
D
${112 cm}^{2}$

Solution:

It is given that

PQ | | BA

PR | | CA

and

PD = 12 cm

.
Basic proportionality theorem states that if a line is drawn parallel to one side of a triangle intersecting the other two sides in distinct points, then the other two sides are divided in the same ratio.
It is given that

PQ | | BA

. Thus,

PQ | | BR

. In

ΔBDR

, by basic proportionality theorem,

\frac{PD}{BD} = \frac{DQ}{DR}

……(1)
As

PR | | CA

so,

PR | | CQ

.
Thus, in

Δ P RD

by basic proportionality theorem,

\frac{CD}{DP} = \frac{DQ}{DR}

……..(2)
From the equations (1) and (2),

\frac{PD}{BD} = \frac{CD}{PD}

Cross multiply the above equation.

BD \times CD = {PD}^{2}

\Rightarrow BD \times CD = 12^{2}

\Rightarrow BD \times CD = 144 {cm}^{2}

Therefore, the value of

BD \times CD

144 {cm}^{2}

.
Hence, option 1 is correct.

In the given figure, $PQ | | BA$ and $PR | | CA$ . If $PD = 12 cm$ then find $BD \times CD$ .

Solution:

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