1 of 3

CIRCLE

  • Sum based on theorem-

Radius is perpendicular to the tangent

2 of 3

Q. In the adjoining figure,

line AB is tangent to both the

circles touching at A and B.

OA = 29, BP = 18, OP = 61

then find AB.

A

B

O

P

A

B

29

18

61

?

To find AB, we need to do a construction

Draw PM ⊥ to seg OA, A-M-O

M

Construction:

Draw seg PM ⊥ seg OA

Sol.

AB belongs to □PBAM

In □PBAM,

∠PBA = 90º

∠BAM = 90º

[Radius is perpendicular

to the tangent]

∠PMA = 90º

[Construction]

∠MPB = 90º

[Remaining Angle]

We know, opposite sides of a rectangle are equal

[By definition]

∴

□PBAM is a rectangle

[Opposite sides of a rectangle]

∴

PB

=

AM

=

18 units

AB = MP

Hint :

Find MP

[ A-M-O]

∴

OA

=

OM

+

AM

29

=

OM

+

18

∴

OM

=

29

–

18

∴

OM

=

11 units

∴

11

∴

…(ii)

∴

AB

=

MP

…(i)

3 of 3

[Opposite sides of a rectangle]

In ΔPMO,

∠PMO = 90º

OP2 =

But, AB = PM

OM2

+ PM2

612

112

+ PM2

3721 =

121

+ PM2

PM2 =

3721

– 121

PM2 =

3600

PM =

60 units

AB =

60 units

OM

=

11 units

Consider ΔPMO

Now, let us apply Pythagoras theorem

=

29

61

?

M

11

∴

∴

∴

∴

∴

∴

Sol.

60

60

Q. In the adjoining figure,

line AB is tangent to both the

circles touching at A and B.

OA = 29, BP = 18, OP = 61

then find AB.

Taking square roots

18

Hint :

Find MP

✔

∴

A

B

O

P

[Pythagoras theorem]