Copyright © Cengage Learning. All rights reserved. Circles 6 6 Chapter.

Slides:



Advertisements
Similar presentations
Secants and Tangents Lesson 10.4 A B T. A B A secant is a line that intersects a circle at exactly two points. (Every secant contains a chord of the circle.)
Advertisements

10.1 Tangents to Circles.
Lesson 6.1 Tangents to Circles
10.4 Secants and Tangents A B T. A B A secant is a line that intersects a circle at exactly two points. (Every secant contains a chord of the circle.)
10.1 Use Properties of Tangents
Lesson 5 Circles.
Tangents, Arcs, and Chords
Unit 25 CIRCLES.
Definitions A circle is the set of all points in a plane that are equidistant from a given point called the center of the circle. Radius – the distance.
10.1 Use Properties of Tangents.  Circle - the set of all points in a plane that are equidistant from a given point.  Center - point in the middle of.
Circle. Circle Circle Tangent Theorem 11-1 If a line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of.
Tangents to Circles Pg 595. Circle the set of all points equidistant from a given point ▫Center Congruent Circles ▫have the same radius “Circle P” or.
Section 12.1: Lines That intersect Circles
By Mark Hatem and Maddie Hines
C HORDS, S ECANTS, AND T ANGENTS 10.1 & S ECANT A secant is a line that intersects a circle in two points. A tangent is a line that intersects the.
CIRCLES Chapter 10.
Circles Chapter 10.
8.1 Circle Terminology and Chord Properties
6.1 Use Properties of Tangents
Tangents to Circles (with Circle Review)
Circles and Chords. Vocabulary A chord is a segment that joins two points of the circle. A diameter is a chord that contains the center of the circle.
Chapter 10.1 Notes: Use Properties of Tangents Goal: You will use properties of a tangent to a circle.
Use Properties of Tangents
Chapter 10.
Chapter 10 Properties of Circles.
 A circle is defined by it’s center and all points equally distant from that center.  You name a circle according to it’s center point.  The radius.
1 Circles. 2 3 Definitions A circle is the set of all points in a plane that are the same distance from a fixed point called the center of the circle.
Lesson 7.3. If the diameter of a circle is 15 units in length, how long is the circle's radius?(answer in a decimal)
8-3 & 8-4 TANGENTS, ARCS & CHORDS
Geometry 11-1 Circle Basics Divide your paper into eight boxes. Draw a circle in each box, but leave some room to write a definition.
Warm Up Directions: Create your own definition for as many of the vocabulary words listed below. You may use diagrams in your explanations. circle radiusdiameter.
Chapter 10 Circles Section 10.1 Goal – To identify lines and segments related to circles To use properties of a tangent to a circle.
Warm-Up Find the area and circumference of a circle with radius r = 4.
Lesson 8-1: Circle Terminology
A radius drawn to a tangent at the point of tangency is perpendicular to the tangent. l C T Line l is tangent to Circle C at point T. CT  l at T.
10.3 – Apply Properties of Chords. In the same circle, or in congruent circles, two ___________ arcs are congruent iff their corresponding __________.
10.1 Tangent Properties to a Circle. POD 1. What measure is needed to find the circumference or area of a circle? 2. Find the radius of a circle with.
Find Segment Lengths in Circles Lesson Definition When two chords intersect in the interior of a circle, each chord is divided into two segments.
Section 9-7 Circles and Lengths of Segments. Theorem 9-11 When two chords intersect inside a circle, the product of the segments of one chord equals the.
10.1 Tangents to Circles. Some definitions you need Circle – set of all points in a plane that are equidistant from a given point called a center of the.
Circle Geometry.
12.1 Parts of Circles and Tangent Lines Geometry.
Copyright © Cengage Learning. All rights reserved. Circles 6 6 Chapter.
10.3 – Apply Properties of Chords
Copyright © 2014 Pearson Education, Inc.
10.5 Segment Lengths in Circles
Chapter 10: Properties of Circles
Chords, secants and tangents
Secants and Tangents A B T.
Module 19: Lesson 4 Segment Relationships in Circles
Circles Lesson 10.1.
Lines that Intersect Circles
Secants and Tangents Lesson 10.4
CIRCLES.
8-3 & 8-4 TANGENTS, ARCS & CHORDS
Section 6.3: Line and Segment Relationships in the Circle.
Copyright © Cengage Learning. All rights reserved.
Day 3.
CIRCLES.
8-6 Segments in Circles.
Learning Target 17 Tangents Lesson 8-3: Tangents.
8-6 Segments in Circles.
Segment Lengths in Circles
Special Segments in a Circle
NOTES 10.4 Secants and Tangents
Copyright © Cengage Learning. All rights reserved.
6.6 Finding Segment Lengths.
Special Segments in a Circle
Segment Lengths in Circles
Special Segments in a Circle
Presentation transcript:

Copyright © Cengage Learning. All rights reserved. Circles 6 6 Chapter

Copyright © Cengage Learning. All rights reserved. Line and Segment Relationships in the Circle 6.3

3 Line and Segment Relationships in the Circle Theorem If a line is drawn through the center of a circle perpendicular to a chord, then it bisects the chord and its arc. Theorem If a line through the center of a circle bisects a chord other than a diameter, then it is perpendicular to the chord.

4 Line and Segment Relationships in the Circle Figure 6.39 (a) illustrates the following theorem. Theorem The perpendicular bisector of a chord contains the center of the circle. Figure 6.39 (a)

5 Example 1 Given: In Figure 6.40, O has a radius of length 5 at B and OB = 3 Find: CD Figure 6.40

6 Example 1 – Solution Draw radius. By the Pythagorean Theorem, (OC) 2 = (OB) 2 + (BC) = (BC) 2 25 = 9 + (BC) 2 (BC) 2 = 16 BC = 4 We know that CD = 2  BC; then it follows that CD = 2  4 = 8.

7 CIRCLES THAT ARE TANGENT

8 Circles that are Tangent In this section, we assume that two circles are coplanar. Although concentric circles do not intersect, they do share a common center. For the concentric circles shown in Figure 6.41, the tangent of the smaller circle is a chord of the larger circle. Figure 6.41

9 Circles that are Tangent If two circles touch at one point, they are tangent circles. In Figure 6.42(a), circles P and Q are internally tangent; in Figure 6.42(b), circles O and R are externally tangent. Figure 6.42 (a) (b)

10 Circles that are Tangent Definition For two circles with different centers, the line of centers is the line (or line segment) containing the centers of both circles. As the definition suggests, the line segment joining the centers of two circles is also commonly called the line of centers of the two circles. In Figure 6.43, or is the line of centers for circles A and B. Figure 6.43

11 COMMON TANGENT LINES TO CIRCLES

12 Common Tangent Lines to Circles A line, line segment, or ray that is tangent to each of two circles is a common tangent for these circles. If the common tangent does not intersect the line segment joining the centers, it is a common external tangent.

13 Common Tangent Lines to Circles In Figure 6.44(a), circles P and Q have one common external tangent, ; in Figure 6.44(b), circles A and B have two common external tangents, and. Figure 6.44 (a) (b)

14 Common Tangent Lines to Circles If the common tangent for two circles does intersect the line of centers for these circles, it is a common internal tangent for the two circles. In Figure 6.45 (a), is a common internal tangent for externally tangent circles O and R; in Figure 6.45(b), and are common internal tangents for M and N. Figure 6.45 (a) (b)

15 Common Tangent Lines to Circles Theorem The tangent segments to a circle from an external point are congruent.

16 Example 2 A belt used in an automobile engine wraps around two pulleys with different lengths of radii. Explain why the straight pieces named and have the same length. Figure 6.47

17 Example 2 – Solution Pulley centered at O has the larger radius length, so we extend and to meet at point E. Because E is an external point to both O and P, we know that EB = ED and EA = EC by Theorem By subtracting equals from equals, EA – EB = EC – ED. Because EA – EB = AB and EC – ED = CD, it follows that AB = CD.

18 LENGTHS OF LINE SEGMENTS IN A CIRCLE

19 Lengths of Line Segments in a Circle To complete this section, we consider three relationships involving the lengths of chords, secants, or tangents. Theorem If two chords intersect within a circle, then the product of the lengths of the segments (parts) of one chord is equal to the product of the lengths of the segments of the other chord.

20 Example 4 In Figure 6.50, HP = 4, PJ = 5, and LP = 8. Find PM. Solution: Applying Theorem 6.3.5, we have HP  PJ = LP  PM. Then 4  5 = 8  PM 8  PM = 20 PM = 2.5 Figure 6.50

21 Lengths of Line Segments in a Circle Theorem If two secant segments are drawn to a circle from an external point, then the products of the length of each secant with the length of its external segment are equal. Theorem If a tangent segment and a secant segment are drawn to a circle from an external point, then the square of the length of the tangent equals the product of the length of the secant with the length of its external segment.