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Question

State true or false:

In ΔABC, angle ABC is equal to twice the angle ACB, and the bisector of angle ABC meets
the opposite side at the point P. Then, AB×BC=BP×CA

A
True
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B
False
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Solution

The correct option is A True
Given: ABC=2ACB, BP bisects ABC
ABC=2ACB
12ABC=ACB
ABP=ACB (AP bisects ACB) (I)
Now, In APB and ABC
ABP=ACB (From I)
BAP=BAC (Common)
APB=ACB (third angle)
thus, APBABC
Hence, ABAC=BPBC...(Corresponding sides)
AB×BC=BP×AC

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