Two Circles Touch each Other

Here we will prove that if two circles touch each other, the point of contact lies on the straight line joining their centres.


Case 1: When the two circles touch each other externally.

Given: Two circles with centres O and P touch each other externally at T.

Two Circles Touch each Other Externally

To prove: T lies on the line OP.

Construction: Draw a common tangent XY through the point of contact T. Join T to O and P.

Proof:

Statement

Reason

1. ∠OTX = 90°

1. Radius OT ⊥ tangent XY.

2. ∠PTX = 90°

2. Radius PT ⊥ tangent XY.

3. ∠OTX + ∠PTX = 180°

⟹ ∠OTP = 180°

⟹ OTP is a straight line

⟹ T lies on OP. (Proved)

3. Adding statement 1 and 2.


Case 2: When the two circles touch each other internally at T.

To prove: T lies on OP produced.

Two Circles Touch each Other Internally

Construction: Draw a common tangent XY through the point of contact T. Join T to O and P.

Proof:

Statement

Reason

1. ∠OTX = 90°

1. Radius OT ⊥ tangent XY.

2. ∠PTX = 90°

2. Radius PT ⊥ tangent XY.

3. OT and PT are both ⊥ to XY at the same point T.

3. From statement 1 and 2.

4. OT and PT lies on the same straight line

⟹ OTP is a straight line

⟹ T lies on OP. (Proved)

4. Only one perpendicular can be drawn to a line through a point on it.

Note: Let two circles with centres O and P touch each other at T. Let OT = r1 and PT = r2 and r1 > r2.


Let the distance between their centres = OP = d.

It is clear from the figures that

• When the circles touch externally, d = r1 + r2.

Circles Touch Externally
Circles Touch Internally

• When the circles touch internally, d = r1 - r2.





10th Grade Math

From Two Circles Touch each Other to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.




Share this page: What’s this?

Recent Articles

  1. Successor and Predecessor | Successor of a Whole Number | Predecessor

    Jul 29, 25 12:59 AM

    Successor and Predecessor
    The number that comes just before a number is called the predecessor. So, the predecessor of a given number is 1 less than the given number. Successor of a given number is 1 more than the given number…

    Read More

  2. Worksheet on Area, Perimeter and Volume | Square, Rectangle, Cube,Cubo

    Jul 28, 25 01:52 PM

    Volume of a Cuboids
    In this worksheet on area perimeter and volume you will get different types of questions on find the perimeter of a rectangle, find the perimeter of a square, find the area of a rectangle, find the ar…

    Read More

  3. Worksheet on Volume of a Cube and Cuboid |The Volume of a RectangleBox

    Jul 25, 25 03:15 AM

    Volume of a Cube and Cuboid
    We will practice the questions given in the worksheet on volume of a cube and cuboid. We know the volume of an object is the amount of space occupied by the object.1. Fill in the blanks:

    Read More

  4. Volume of a Cuboid | Volume of Cuboid Formula | How to Find the Volume

    Jul 24, 25 03:46 PM

    Volume of Cuboid
    Cuboid is a solid box whose every surface is a rectangle of same area or different areas. A cuboid will have a length, breadth and height. Hence we can conclude that volume is 3 dimensional. To measur…

    Read More

  5. Volume of a Cube | How to Calculate the Volume of a Cube? | Examples

    Jul 23, 25 11:37 AM

    Volume of a Cube
    A cube is a solid box whose every surface is a square of same area. Take an empty box with open top in the shape of a cube whose each edge is 2 cm. Now fit cubes of edges 1 cm in it. From the figure i…

    Read More