CIRCLE
Theorem – Radius is perpendicular
to the tangent
Q. There are two concentric circles with O centre of radii 5 cm
and 3 cm. from an external point P, tangents PA and PB are
drawn to these circles. If AP = 12cm, find the length of BP.
A
O
B
P
Sol.
∠OAP = 90º
OP2
=
OA2
+
AP2
∴
OP2
=
52
+
122
∴
OP2
=
25
+
144
[By Pythagoras theorem]
[radius is perpendicular to tangent]
In ΔOAP,
∴
OP2
=
169
∴
OP
=
13
∠OBP = 900
OB2
+
BP2
=
OP2
∴
32
+
BP2
=
132
∴
BP2
=
169
–
9
[By Pythagoras theorem]
[radius is perpendicular to tangent]
In ΔOBP,
∴
BP2
=
160
∴
BP
=
160
∴
BP
=
16
×
10
∴
BP
=
4
10
cm
12cm
?
Now, let us apply Pythagoras theorem
Consider ΔOAP
5cm
13cm
Now, let us apply Pythagoras theorem
consider ΔOBP
3cm
Observe ∠OAP
∴∠OAP = 90º
We know that, radius is perpendicular to the tangent
Observe ∠OBP
∴∠OBP = 90º
We know that, radius is perpendicular to the tangent
[Taking square-roots]