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

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80 cm

76 cm

72 cm

64 cm

answer is C.

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Detailed Solution

Let the diagram of the given be,
Question Image

Assume that P is appointed somewhere on side BC,

P A = 2 P D

… (1) and the length of the BP be x. So,

P C = 90 − x

.
To find the value of x, we have to use Pythagoras theorem.
In right-angled triangle ABP, we get

P A 2 = A B 2 + B P 2 P A 2 = 362 + x 2

Similarly, in right-angled triangle CPD, we have

P D 2 = P C 2 + D C 2 P D 2 = 362 + (90 − x) 2

Substitute the above values in (1), and we have

P A = 2 P D P A 2 = 4 P D 2 362 + x 2 = 4 (362 + (90 − x) 2) x 2 = 3 × 362 + [4 × (8100 + x 2 − 180 x)] x 2 = 3888 + 32400 + 4 x 2 − 720 x 3 x 2 − 720 x + 36288 = 0 x 2 − 240 x + 12096 = 0

By the factorization, we get

(x − 168) (x − 72) = 0 x = 168, 72

Since

168 > 90

is not possible, then the length is 72 cm.
Hence, the correct option is 3.

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