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MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. What is Unit 3 about? We will learn to use arcs, angles, and segments in circles to solve real life problems. We will learn how to find the measure of angles related to a circle. We will find circumference and area of figures with circles. We will also find the surface and volume of spheres.

MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle

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MM2G3 Students will understand properties of circles.

MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.

What is Unit 3 about?

We will learn to use arcs, angles, and segments in circles to solve real life problems. We will learn how to find the measure of angles related to a circle. We will find circumference and area of figures with circles. We will also find the surface and volume of spheres.

MM2G3 Students will understand properties of circles.

MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.

Crop Circles

http://www.coolmath.com/lesson-geometry-of-crop-circles-1.html

Whether you think crop circles are made by little green men from space or by sneaky earthling geeks, you've got to admit that they are pretty dang cool...  And whoever is making them knows a ton of geometry!

MM2G3 Students will understand properties of circles.

MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.

Properties of Tangents

Essential Question:

How do we identify segments and lines related to circles and how do we use properties of a tangent to a circle?

M2 Unit 3: Day 1

Tuesday, April 18, 2023

Lesson 6.1

MM2G3 Students will understand properties of circles.

MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.

1. What measure is needed to find the circumferenceor area of a circle?

2. Find the radius of a circle with diameter 8 centimeters.

3. A right triangle has legs with lengths 5 inches and12 inches. Find the length of the hypotenuse.

ANSWER 13 in.

4. Solve 6x + 15 = 33. 5. Solve (x + 18)2 = x2 + 242.

ANSWER 7

Warm Ups

MM2G3 Students will understand properties of circles.

MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.

MM2G3 Students will understand properties of circles.

MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.

Circle

AC

Secant

D

E

Chord

FTangent

P

Name of the circle: ʘ P

Diameter Radius

B

Circle The set of all points in a plane that are equidistant from a given point, called the center.

MM2G3 Students will understand properties of circles.

MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.

Definition

Diameter – a chord that contains the center of the circle.

diameter

radiusP

Radius – a segment from the center of the circle to any point on the circle.

is a radiusPC

is a diameterABA B

C

MM2G3 Students will understand properties of circles.

MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.

Definition Chord – a segment whose endpoints are points on the

circle.

AB is a chord

B

A

MM2G3 Students will understand properties of circles.

MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.

Definition Secant – a line that intersects a circle in two points.

MN is a secant

N

M

MM2G3 Students will understand properties of circles.

MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.

Definition Tangent – a line in the plane of a circle that intersects

the circle in exactly one point.

ST is a tangent

S

T

O

P

Point of tangency

MM2G3 Students will understand properties of circles.

MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.

A little extra information

♥ The word tangent comes from the Latin word meaning to touch

♥ The word secant comes from the Latin word meaning to cut.

MM2G3 Students will understand properties of circles.

MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.

1. Tell whether the line or segment is best described as a chord, a secant, a tangent, a diameter, or a radius.

FC

B

G

A

H

D

E

Id. CE

c. DF

b. EI

a. AH tangent

diameter

chord

radius

EXAMPLE 1

MM2G3 Students will understand properties of circles.

MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.

EXAMPLE 2 Identify special segments and lines

2. Tell whether the line, ray, or segment is best described as a radius, chord, diameter, secant, or tangent of ʘ C.

ACa.

SOLUTION

is a radius because C is the center and A is a point on the circle.

AC

a.

b. AB is a diameter because it is a chord that contains the center C.

b. AB

c. DE is a tangent ray because it is contained in a line that intersects the circle at only one point.

DEc.

d. AE is a secant because it is a line that intersects the circle in two points.

AEd.

MM2G3 Students will understand properties of circles.

MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.

EXAMPLE 3 Find lengths in circles in a coordinate plane

b. Diameter of ʘ A

Radius of ʘ Bc.

Diameter of ʘ Bd.

3. Use the diagram to find the given lengths.

a. Radius of ʘ A

SOLUTION

a. The radius of ʘ A is 3 units.

b. The diameter of ʘ A is 6 units.

c. The radius of ʘ B is 2 units.

d. The diameter of ʘ B is 4 units.

MM2G3 Students will understand properties of circles.

MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.

SOLUTION

GUIDED PRACTICE

a. The radius of ʘ C is 3 units.

b. The diameter of ʘ C is 6 units.

c. The radius of ʘ D is 2 units.

d. The diameter of ʘ D is 4 units.

4. Find the radius and diameter of ʘ C and ʘ D.

EXAMPLE 4

MM2G3 Students will understand properties of circles.

MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.

Definitions Common tangent – a line or segment that is tangent to two

coplanar circles Common internal tangent – intersects the segment that joins the

centers of the two circles Common external tangent – does not intersect the segment that joins

the centers of the two circles

common external tangentcommon internal tangent

MM2G3 Students will understand properties of circles.

MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.

5. Tell whether the common tangents are internal or external.

a. b.

common internal tangents common external tangents

EXAMPLE 5

MM2G3 Students will understand properties of circles.

MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.

EXAMPLE 6 Draw common tangents

6. Tell how many common tangents the circles have and draw them.

a. b. c.

SOLUTION

a. 4 common tangents 3 common tangentsb.

c. 2 common tangents

MM2G3 Students will understand properties of circles.

MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.

Perpendicular Tangent Theorem 6.1

In a plane, if a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.

If l is tangent to Q at P, then l QP.

l

Q

P

MM2G3 Students will understand properties of circles.

MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.

Tangent Theorems

Create right triangles for problem solving.

MM2G3 Students will understand properties of circles.

MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.

EXAMPLE 8 Find the radius of a circle

7. In the diagram, B is a point of tangency. Find the radius r of ʘC.

SOLUTION

You know that AB BC , so △ ABC is a right triangle. You can use the Pythagorean Theorem.

AC2 = BC2 + AB2

(r + 50)2 = r2 + 802

r2 + 100r + 2500 = r2 + 6400100r = 3900

r = 39 ft .

Pythagorean Theorem

Substitute.

Write the binomial twice.

Subtract from each side.

Divide each side by 100.

(r + 50)(r + 50) = r2 + 802

r2 + 50r +50r + 2500 = r2 + 6400

Combine Like Terms.

Multiply.

MM2G3 Students will understand properties of circles.

MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.

8. ST is tangent to ʘ Q. Find the value of r.

SOLUTION

You know from Theorem 10.1 that ST QS , so △ QST is a right triangle. You can use the Pythagorean Theorem.

r2 + 36r + 324 = r2 + 576

36r = 252

r = 7

Multiply.

Subtract from each side.

Divide each side by 36.

QT2 = QS2 + ST2

(r + 18)2 = r2 + 242

Pythagorean Theorem

Substitute.

EXAMPLE 8

MM2G3 Students will understand properties of circles.

MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.

Perpendicular Tangent Converse

In a plane, if a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is tangent to the circle.

l

Q

P

If l QP at P, then l is tangent to Q.

MM2G3 Students will understand properties of circles.

MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.

GUIDED PRACTICE

9. Is DE tangent to ʘ C?

ANSWER

Yes – The length of CE is 5 because the radius is 3 and the outside portion is 2. That makes ∆CDE a 3-4-5 Right Triangle. So DE and CD are

EXAMPLE 10

MM2G3 Students will understand properties of circles.

MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.

EXAMPLE 7 Verify a tangent to a circle

SOLUTION

Use the Converse of the Pythagorean Theorem. Because 122 + 352 = 372, △ PST is a right triangle and ST PT . So, ST is perpendicular to a radius of ʘ P at its endpoint on ʘ P. ST is tangent to ʘ P.

10. In the diagram, PT is a radius of ʘ P. Is ST tangent to ʘ P ?

MM2G3 Students will understand properties of circles.

MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.

Congruent Tangent Segments Theorem 6.2

If two segments from the same exterior point are tangent to a circle, then they are congruent.

SP

R

T

If SR and ST are tangent to P, then SR ST.

MM2G3 Students will understand properties of circles.

MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.

RS is tangent to ʘ C at S and RT is tangent to ʘC at T.

Find the value of x.

SOLUTION

RS = RT

28 = 3x + 4

8 = x

Substitute.

Solve for x.

Tangent segments from the same point are

11.

MM2G3 Students will understand properties of circles.

MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.

AB is tangent to C at B.AD is tangent to C at D.

Find the value of x.

11

x2 + 2

AC

D

BAD = AB

x2 + 2 = 11

x2 = 9

x = 3

12.

MM2G3 Students will understand properties of circles.

MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.

HomeworkPage 187 # 18 – 24 all Page 188 # 1 – 10.