Processing math: 100%

Problems on Common Tangents to Two Circles

Here we will solve different types of  problems on common tangents to two circles.

1. There are two circles touch each other externally. Radius of the first circle with centre O is 8 cm. Radius of the second circle with centre A is 4 cm. Find the length of their common tangent BC.

Common Tangent to Two Circles

Solution:

Join O to A and B. Join A to C. Draw DA ⊥ OB.

Now DA = BC, as they are opposite sides of rectangle ACBD.

OA = 8 cm + 4 cm

     = 12 cm.

OD = 8 cm – 4cm

     = 4 cm.

Therefore, DA = √OA2−OD2

                     = √122−42 cm

                     = √144−16 cm

                     = √128 cm

                     = 8√2 cm

Therefore, BC = 8√2 cm.


2. Prove that a transverse common tangent drawn to two circles divides the line joining their centres into the ratio of their radii.

Problems on Transverse Common Tangents

Solution:

Given: Two circles with centres O and P, and radii OX and PY respectively. The transverse common tangent XY touches them at X and Y respectively. XY cuts OP at T.

To prove: OTTP = OXPY.

Proof:

Statement

Reason

1. In ∆XOT and ∆YPT,

(i) ∠OXT = ∠PYT = 90 °

(ii) ∠OTX = ∠PTY.

1.

(i) Tangent ⊥ Radius.

(ii) Vertically opposite angles.

2. ∆XOT ∼ ∆YPT

2. By A – A criterion of similarity.

3. Therefore, OTTP = OXPY. (Proved)

3. Corresponding sides of similar triangles are proportional.






10th Grade Math

From Problems on Common Tangents to Two Circles 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

RSS

  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