# Geometry (Length of common tangent between two circles)

### 1. If two circles of radius $r_{1}$ and $r_{2}$ touch each other externally, then the length of the direct common tangent is $\sqrt{4r_{1}r_{2}}$

Proof : Let the length of the common tangent be l

Draw a line OR parallel to PQ

$\angle$OPQ = 90

$\angle$O’QP = 90

{ line joining the center of the circle to the point of contact makes an angle of 90 degree with the tangent }

$\angle$OPQ + $\angle$O’QP = 180

OP $\parallel$ QR

Since opposite sides are parallel and interior angles are 90, therefore OPQR is a rectangle.

So OP = QR = $r_{1}$   and PQ = OR = l

In $\triangle$OO’R

$\angle$ORO’ = 90

By Pythagoras theorem

$OR^{2}$$O'R^{2}$$OO'^{2}$ $l^{2}$$(r_{1}-r_{2})^{2}$$(r_{1}+r_{2})^{2}$ $l^{2}$$r_{1}^{2}+r_{2}^{2}-2r_{1}r_{2}$$r_{1}^{2}+r_{2}^{2}+2r_{1}r_{2}$ $l^{2}$$4r_{1}r_{2}$

l = $\sqrt{4r_{1}r_{2}}$

### 2. There are two circle of radius $r_{1}$ and $r_{2}$ which intersect each other at two points. If their centers are d units apart , then the length of the direct common tangent between them is $\sqrt{d^{2}-(r_{1}-r_{2})^{2}}$

Proof Let PQ = l

Draw a line OR parallel to PQ

$\angle$OPQ = 90

$\angle$O’QP = 90

{ line joining the center of the circle to the point of contact makes an angle of 90 degree with the tangent }

$\angle$OPQ + $\angle$O’QP = 180

OP $\parallel$ QR

Since opposite sides are parallel and interior angles are 90, therefore OPQR is a rectangle.

So OP = QR = $r_{1}$   and PQ = OR = l

In $\triangle$OO’R

$\angle$ORO’ = 90

By Pythagoras theorem

$OR^{2}$$O'R^{2}$$OO'^{2}$ $l^{2}$$(r_{1}-r_{2})^{2}$$d^{2}$ $l^{2}$  = $d^{2}-(r_{1}-r_{2})^{2}$

l = $\sqrt{d^{2}-(r_{1}-r_{2})^{2}}$

### 3. If the centers of two circle of radius $r_{1}$ and $r_{2}$  are d units apart , then the length of the direct common tangent between them is $\sqrt{d^{2}-(r_{1}-r_{2})^{2}}$

Draw a line OR parallel to PQ

$\angle$OPQ = 90

$\angle$O’QP = 90

{ line joining the center of the circle to the point of contact makes an angle of 90 degree with the tangent }

$\angle$OPQ + $\angle$O’QP = 180

OP $\parallel$ QR

Since opposite sides are parallel and interior angles are 90, therefore OPQR is a rectangle.

So OP = QR = $r_{1}$   and PQ = OR = l

In $\triangle$OO’R

$\angle$ORO’ = 90

By Pythagoras theorem

$OR^{2}$$O'R^{2}$$OO'^{2}$ $l^{2}$$(r_{1}-r_{2})^{2}$$d^{2}$ $l^{2}$  = $d^{2}-(r_{1}-r_{2})^{2}$

l = $\sqrt{d^{2}-(r_{1}-r_{2})^{2}}$

### 4. If the centers of two circle of radius $r_{1}$ and $r_{2}$  are d units apart , then the length of the transverse common tangent between them is $\sqrt{d^{2}-(r_{1}+r_{2})^{2}}$

Draw a line O’R parallel to PQ and extend OP to PR as shown in the figure

$\angle$OPQ = 90

$\angle$O’QP = 90

{ line joining the center of the circle to the point of contact makes an angle of 90 degree with the tangent }

$\angle$OPQ + $\angle$O’QP = 180

OP $\parallel$ QR

Since opposite sides are parallel and interior angles are 90, therefore OPQR is a rectangle.

So O,P = RP = $r_{2}$   and PQ = O’R = l

In $\triangle$OO’R

$\angle$ORO’ = 90

By Pythagoras theorem

$O'R^{2}$$OR^{2}$$OO'^{2}$ $l^{2}$$(r_{1}+r_{2})^{2}$$d^{2}$ $l^{2}$  = $d^{2}-(r_{1}+r_{2})^{2}$

l = $\sqrt{d^{2}-(r_{1}+r_{2})^{2}}$

### Problems for practise

1. Two circles touch each other externally and the center of two circles are 13 cm apart. If the length of the direct common tangent between them is 12 cm, find the radius of the bigger circle

a) 6 cm                   b) 8 cm                      c) 9 cm                     d) 5 cm

2.  The center of two circles of radius 5 cm and 3 cm are 17 cm apart . Find the length of the transverse common tangent between them

a) 15 cm                  b) 12 cm                       c) 10 cm                      d) 9 cm

3.The center of two circles are 10 cm apart and  the length of the direct common tangent between them is approximate 9.5 cm. If the radius of one circle is 4 cm , find the radius of another circle

a) 5 cm                         b) 1 cm                          c) 7 cm                           d) 3 cm

4. There are two circles which do not touch or intersect each other. If the radius of two circles are 7 cm and 5 cm respectively and the length of the transverse common tangent between them is 9 cm , find the distance between their centers

a)10 cm                 b) 20 cm                       c) 12 cm                                 d) 15 cm

5. Two circles of radius 8 cm and 5 cm intersect each other at two points A and B. If the distance between their centers is 5 cm, find the length of the direct common tangent between them

a) 3 cm                    b) 4 cm                        c) 6 cm                               d) 2 cm