**Problem 1 :**

A circle with a radius of 8 cm is externally tangent to a circle with a radius of 18 cm. Find the length of the external tangent.

**Solution :**

When we join the point from Q on the side AP, we get a rectangle ABCQ and triangle CPQ.

In triangle CPQ :

PQ = 18+8 ==> 26

PQ^{2} = CP^{2} + CQ^{2}

26^{2} = 10^{2} + CQ^{2}

676 - 100 = CQ^{2}

CQ^{2} = 576

CQ = √576

CQ = 24

CQ = AB = 24 cm

So, length of external tangent is 24 cm.

**Problem 2 :**

The center of two circles of radii 10 cm and 5 cm are 13 cm apart.

(a) Find the length of external tangent.

(b) Do the circles intersect.

**Solution :**

Let us draw a rough picture.

The picture drawn above does not satisfy the condition. Because in the picture given above, the length of PQ is 15 but according to the given information the length between their centers is 13.

So, we draw the picture

In triangle CPQ :

PQ = 13 (given)

PQ^{2} = CP^{2} + CQ^{2}

13^{2} = 5^{2} + CQ^{2}

169 - 25 = CQ^{2}

CQ^{2} = 144

CQ = √144

CQ = 12

CQ = AB = 12 cm

So, length of external tangent is 12 cm.

(b) Yes, both circles are touching at 2 points.

**Problem 3 :**

RD is a tangent to a circle A and H. HA = 40 cm, the radius of circle A is 8 cm and radius of circle H is 12 cm. Find the length of RD.

**Solution :**

When we draw a line from D and that is parallel to RH, it becomes the shape rectangle RHDC.

Triangle AHC is a right triangle.

<ACH = 90 degree

AC = AD + DC ==> 12, RH = 8and AH = 40

Using Pythagorean theorem, we get

AH^{2} = AC^{2} + CH^{2}

40^{2} = 12^{2} + CH^{2}

1600 - 144 = CH^{2}

CH^{2} = 1456

CH = √1456

CH = 38.15

RD = 38.15

So, the length of external tangent is 38.15 cm.

**Problem 4 :**

AB is tangent to both circles. Find the value of x and the distance between the centers.

Solution :

Triangle CEA is right triangle.

CE^{2} = AC^{2} + AE^{2}

CE^{2} = 5^{2} + 8^{2}

CE^{2} = 89

CE = √89

In triangle CEA and triangle EBD

<CAE = <BDE (A)

<CEA = <BED (A)

Using AA, triangles CEA and EBD are similar.

So, their corresponding sides will be the same ratio.

CE/ED = CA/BD = AE/EB

√89/ED = 5/x = 8/12

5/x = 8/12

x = 60/8

x = 7.5

√89/ED = 5/x

By applying the value of x above, we get

√89/ED = 5/7.5

9.4(7.5)/5 = ED

ED = 14.1

Distance between centers = CD

= CE + ED

= 9.4+14.1

= 23.5

So, distance between two centers is 23.5 cm.

**Problem 5 :**

Two circles that are externally tangent have radii 12 inches and 8 inches respectively. Find the length of the tangent AB.

**Solution :**

DC = 8 inches and let OD = x.

In triangles AQB and AOC,

<QAB = <OAC

<ABQ = <ACO (90)

Using AA,

ΔAQB ~ ΔAOC

Corresponding sides will be in the same ratio.

OB/OC = AQ/AO = AB/AC

8/12 = AQ/(20+AQ) = AB/AC

2/3 = AQ/(20+AQ)

2(20+AQ) = 3AQ

AQ = 40

In triangle AQB, using Pythagorean theorem

QA^{2} = OB^{2} + AB^{2}

40^{2} = 8^{2} + AB^{2}

1600-64 = AB^{2}

AB = √1536

AB = 39.2

So, length of tangent AB is 39.2 cm.

Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.

If you have any feedback about our math content, please mail us :

**v4formath@gmail.com**

We always appreciate your feedback.

You can also visit the following web pages on different stuff in math.

**WORD PROBLEMS**

**Word problems on simple equations **

**Word problems on linear equations **

**Word problems on quadratic equations**

**Area and perimeter word problems**

**Word problems on direct variation and inverse variation **

**Word problems on comparing rates**

**Converting customary units word problems **

**Converting metric units word problems**

**Word problems on simple interest**

**Word problems on compound interest**

**Word problems on types of angles **

**Complementary and supplementary angles word problems**

**Trigonometry word problems**

**Markup and markdown word problems **

**Word problems on mixed fractrions**

**One step equation word problems**

**Linear inequalities word problems**

**Ratio and proportion word problems**

**Word problems on sets and venn diagrams**

**Pythagorean theorem word problems**

**Percent of a number word problems**

**Word problems on constant speed**

**Word problems on average speed **

**Word problems on sum of the angles of a triangle is 180 degree**

**OTHER TOPICS **

**Time, speed and distance shortcuts**

**Ratio and proportion shortcuts**

**Domain and range of rational functions**

**Domain and range of rational functions with holes**

**Graphing rational functions with holes**

**Converting repeating decimals in to fractions**

**Decimal representation of rational numbers**

**Finding square root using long division**

**L.C.M method to solve time and work problems**

**Translating the word problems in to algebraic expressions**

**Remainder when 2 power 256 is divided by 17**

**Remainder when 17 power 23 is divided by 16**

**Sum of all three digit numbers divisible by 6**

**Sum of all three digit numbers divisible by 7**

**Sum of all three digit numbers divisible by 8**

**Sum of all three digit numbers formed using 1, 3, 4**

**Sum of all three four digit numbers formed with non zero digits**