Question

ABCD is a rectangle, sides 36 cm and 90 cm. P is a point on BC which is one of the longer sides such that PA = 2PD. The length of PB is

(a) 80 cm

(b) 18 cm

(c) 72 cm

(d) 64 cm

Answer 1

Correct option is (c) 72 cm

ABCD is a rectangle.

So, ΔABP & ΔDPC are right ones with PA & PD as hypotenuses .

PA = 2PD  i.e

PA² = (2PD)² = 4PD²

Let PB = x,  i.e  PC = 90 − x.

So, applying Pythagoras theorem,

PA² = x² + 36²

⇒ 4PD² = x² + 36²  ........(i)

PD² = (90 − x)² + 36² 

⇒ 4PD² = 4(90 − x)² + 4 × 362   ......(ii)

Comparing (i) & (ii),

x² + 36² = 4(90 − x)² + 4 × 362 

⇒ 37584 − 720x + 36288 = 0 

⇒ (x − 168)(x − 72) = 0

⇒ x = (168, 72) cm.

We reject x = PB = 168 as PB is the part of BC = 90 cm.

∴ x = PB = 72cm.