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 = \(\sqrt{OA^{2} - OD^{2}}\)

                     = \(\sqrt{12^{2} - 4^{2}}\) cm

                     = \(\sqrt{144  - 16}\) cm

                     = \(\sqrt{128}\) 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: \(\frac{OT}{TP}\) = \(\frac{OX}{PY}\).

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, \(\frac{OT}{TP}\) = \(\frac{OX}{PY}\). (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. Multiplying 3-Digit Number by 1-Digit Number | Three-Digit Multiplicat

    Oct 22, 24 03:26 PM

    Multiplying 3-Digit Number by 1-Digit Number
    Here we will learn multiplying 3-digit number by 1-digit number. In two different ways we will learn to multiply a two-digit number by a one-digit number. 1. Multiply 201 by 3 Step I: Arrange the numb…

    Read More

  2. Word Problems on Multiplication |Multiplication Word Problem Worksheet

    Oct 22, 24 01:23 AM

    Multiplication Word Problem
    Word problems on multiplication for fourth grade students are solved here step by step. Problem Sums Involving Multiplication: 1. 24 folders each has 56 sheets of paper inside them. How many sheets of…

    Read More

  3. Worksheet on Word Problems on Multiplication | Multiplication Problems

    Oct 22, 24 12:31 AM

    In worksheet on word problems on multiplication, all grade students can practice the questions on word problems involving multiplication. This exercise sheet on word problems on multiplication

    Read More

  4. Multiplying 2-Digit Number by 1-Digit Number | Multiply Two-Digit Numb

    Oct 21, 24 03:38 PM

    Multiplying 2-Digit Number by 1-Digit Number
    Here we will learn multiplying 2-digit number by 1-digit number. In two different ways we will learn to multiply a two-digit number by a one-digit number. Examples of multiplying 2-digit number by

    Read More

  5. Multiplication Table of 4 |Read and Write the Table of 4|4 Times Table

    Oct 21, 24 02:26 AM

    Multiplication Table of Four
    Repeated addition by 4’s means the multiplication table of 4. (i) When 5 candle-stands having four candles each. By repeated addition we can show 4 + 4 + 4 + 4 + 4 = 20 Then, four 5 times

    Read More