In the given figure PQ is parallel to BA and PR is parallel to CA. If PD = 12cm. Find BD × CD.

Given: PQ || BA, PR || CA, PD = 12 cm

To find: BD × CD

Concept Used: Basic Proportionality Theorem

Basic Proportionality Theorem: If a line is parallel to a side of a triangle which intersects the other sides into two distinct points, then the line divides those sides in proportion.

In ∆ BRD, PQ || BR as PQ || BA

So by basic Proportionality Theorem,

……(i)

In ∆ PRD, CQ || PR as PR || CA

So By basic proportionality theorem.

……(ii)

From eq (i) and eq (ii)

Cross multiplying we get,

PD × PD = BD × CD

⇒ BD × CD = PD² = 12²

⇒ BD × CD = 144

∴ the product of BD and CD is 144.