Geometry: An Interactive Journey to Mastery

Lesson 19: The Geometry of a Circle

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x radius/tangent theorem:

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x two tangents theorem:

Two tangent segments from a given point to a circle are congruent.

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x inscribed/central angle theorem:

All inscribed angles from a common arc are congruent and have

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Summary

Having skirted around the topic of circles for a number of lessons, it is

time to study circles as objects of interest in their own right and prove

theorems about them. In this lesson, we establish three classic results: the

radius/tangent theorem, the two tangents theorem, and the inscribed/central

angle theorem. Our work begins with a puzzle that happens to establish the

converse of Thales’s theorem from Lesson 1.

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Find the values of x and y in the )LJXUH.

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By the two tangents theorem, y = 2 and 10 = 3 + x, giving x = 7.

Figure 19.1

Figure 19.2

x

x x

x

x

2 x

Figure 19.3

3

2

10

x

y

Figure 19.4