Points P and Q lie on sides AC and BC of a ΔABC respectively with AB = 20 cm, BC = 12 cm and AC = 10 cm. PY is parallel to AQ and QX is parallel to BP such that Y lies on BC and X on AC. If A(ΔCXY) = 7.2 sq. cm, what is ℓ(XY)?

 

Points P and Q lie on sides AC and BC of a ΔABC respectively with AB = 20 cm, BC = 12 cm and AC = 10 cm. PY is parallel to AQ and QX is parallel to BP such that Y lies on BC and X on AC. If A(ΔCXY) = 7.2 sq. cm, what is ℓ(XY)?

 
Question of the Day (09-Apr-12)


As AQ || PY, we can see that

ΔCPY ~ ΔCAQ

 

As BP || QX, we can see that

ΔCPB ~ ΔCXQ

 

From (i) and (ii),

CP × CQ = CY × CA = CB × CX

 

As a line parallel to the base of the triangle divides the other two sides in equal ratios, we can say that XY || AB.

∴ ΔCXY ~ ΔCAB

 

Semiperimeter, s of ΔABC = (20 + 10 + 12)/2 = 21 cm

 

∴ From (iii),

(XY)² = 7.2 × 400/45 = 64

∴ XY = 8 cm