Question

In a circle AB is the longest chord. P is a point on the circle such that PA=PB. What is the

measure of ∠ABP?


(A) 90° (B) 45° (C) 180° (D) 25°


Class 9

Answer 1

Answer:

Consider a circle C with center O.

We have PA and PB are tangents of the circle, PA=10cm and

∠APB=60

o

Join OP,

Such that,

In △PAC and △PBC we have,

PA=PB [tangent of the circle fro the outer point p is equal]

∠PAC=∠PBC [angle made by the external tangent on a

circle is equal]

PC=CP [common]

so,

△PAC ≅ △PBC [By SAS criteria]

so,

AC=BC (i)

∠ACP=∠BCP .(ii)

since

∠APB=∠APC+∠BPC

so,

∠APC=

2

1

×60

0

=30

0

∠APC=30

0

∠ACP+∠BCP=180

0

from equation 2 we get

∠ACP=

2

1

×180

0

=90

0

Thus in Right △ ACP

sin30

0

=

AP

AC

2

1

=

10

AC

cm

AC=5cm

Since

AC=BC

so,

AB=AC+BC

=5cm+5cm

=10cm