# 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.

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.

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.

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.

## Recent Articles

1. ### Decimal Place Value Chart |Tenths Place |Hundredths Place |Thousandths

Jul 19, 24 03:26 PM

Decimal place value chart are discussed here: The first place after the decimal is got by dividing the number by 10; it is called the tenths place.

2. ### Definition of Decimal Numbers | Decimal Part | Decimal Point |Examples

Jul 19, 24 11:13 AM

Definition of decimal numbers: We have learnt that the decimals are an extension of our number system. We also know that decimals can be considered as fractions whose denominators are 10, 100, 1000

3. ### Addition and Subtraction of Fractions | Solved Examples | Worksheet

Jul 19, 24 02:00 AM

Addition and subtraction of fractions are discussed here with examples. To add or subtract two or more fractions, proceed as under: (i) Convert the mixed fractions (if any.) or natural numbers

4. ### Fractions in Descending Order |Arranging Fractions an Descending Order

Jul 19, 24 02:00 AM

We will discuss here how to arrange the fractions in descending order. Solved examples for arranging in descending order: 1. Arrange the following fractions 5/6, 7/10, 11/20 in descending order. First…