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Elementary Geometry for College St...

6th Edition
Daniel C. Alexander + 1 other
ISBN: 9781285195698

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BuyFindarrow_forward

Elementary Geometry for College St...

6th Edition
Daniel C. Alexander + 1 other
ISBN: 9781285195698
Textbook Problem
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The center of a circle of radius 3 inches is at a distance of 20 inches from the center of a circle of radius 9 inches. What is the exact length of common internal tangent A B ¯ ?

(HINT: Use similar triangles to find OD and DP. Then apply the Pythagorean Theorem twice.)

Chapter 6.3, Problem 38E, The center of a circle of radius 3 inches is at a distance of 20 inches from the center of a circle

To determine

To find:

The find value of AB¯.

Explanation

Given that, the center of a circle of radius 3 in. is at a distance of 20 in. from the center of a circle of radius 9 in.

That is OA=3,PB=9 and OP=20 then AB¯ is common internal tangent.

The diagrammatic representation is given below,

Since AB¯ is common internal tangent therefore, AB¯OA¯ and AB¯PB¯

Using the vertical angle theorem to get the following,

mADO=mPDBmOAD=mPBD

Therefore, mOADmPBD

If two triangles are similar, then the ration of any two corresponding segments (such as altitudes, medians, or angle bisectors) equals the ratio of any two corresponding sides.

OA¯PB¯=OD¯PD¯39=OD¯PD¯OD¯PD¯=133OD¯=PD¯

We know that,

OP¯=OD¯+PD¯20=OD¯+PD¯PD¯=20OD¯. Since OP¯=20

Substitute this in the above equation to get the following,

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