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Secants and Tangents - Lesson 10-4

Today we began by looking at the definition of chords, secants, and tangents of a circle:

Definition of a Chord and a Secant

Definition of a Tangent to a Circle

We then discovered a property of tangents:

Postulates related to Tangents

Next, we defined tangent segments and secant segments:

Definitions of Tangent Segments and Secant Segments

Next, we learned a property of tangents:

The Two Tangents Theorem

We then covered a number of additional definitions:

Definition of a Common External Tangent

Definition of a Common Internal Tangent

Definition of Externally Tangent Circles

Definition of Internally Tangent Circles

We finished by talking about the Common Tangent Procedure (detailed below) and doing the following example...make sure you can follow these steps for either a common internal or common external tangent!

Common Tangent Procedure

The problem was to find x in the diagram below:


Step 1 is to draw in the segment connecting the centers of the circles (shown in red below):

Example Step 1

Step 2 is to draw in radii to the points of tangency of the common tangent. These were already drawn for our example (segments AP and BQ), but for now, let's at least note that they form right angles with the common tangent segment.

Example Step 2

Step 3 is to use the parallel postulate to draw in a line parallel to the tangent through the center of one of our circles (segment RQ). Note that ARQB is a rectangle, so its opposite sides are congruent. Using this fact and a little simple math, we can see that RP = 10.

Example Step 3

Step 4, the last step, uses the Pythagorean Theorem (or a family of right triangles - in this case we have a 5-12-13 triangle), solve for the length of segment RQ, which is the same as x!!

Example Step 4

Note that this procedure (Steps 1 - 4) will also work for finding the length of a common internal tangent. The picture will be different, but the result will be the same!

Other Links
Class Notes
Lesson 10-1
Lesson 10-2
Lesson 10-3
Lesson 10-4
Lesson 10-5
Quiz Topics
Lesson 10-6
Lesson 10-7
Lesson 10-8
Lesson 10-9
Test Topics
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