tangents and chords activity 4.1 off on a tangent...

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Unit 4 • Circles and Constructions 277

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My Notes

ACTIVITY

4.1Tangents and ChordsOff On a TangentSUGGESTED LEARNING STRATEGIES: Group Presentation, Think/Pair/Share, Quickwrite, Interactive Word Wall, Vocabulary Organizer, Create Representations, Quickwrite

A circle is the set of all points in a plane at a given distance from a given point in the plane. Lines and segments that intersect the circle have special names.

! e following illustrate tangent lines to a circle.

1. On the circle below, draw three unique examples of lines or segments that are not tangent to the circle.

2. Write a description of tangent lines.

3. Using the circle below,

a. Draw a tangent line and a radius to the point of tangency.

b. Describe the relationship between the tangent line and the radius of the circle drawn to the point of tangency.

ACADEMIC VOCABULARY

circle

The radius of a circle is a segment, or length of a segment, from the center to any point on the circle

MATH TERMS

A secant is a line that intersects the circle in two points.

MATH TERMS

ACADEMIC VOCABULARY

A tangent is a line in the plane of a circle that intersects the circle at just one point, called the point of tangency.

Answers may vary. Sample answer:

A line is tangent to a circle if and only if it intersects the circle in exactly one point.


A tangent line is perpendicular to a radius that is drawn to the point of tangency.

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ACTIVITY 4.1 Investigative

Tangents and Chords

Activity Focus• Relationships among tangent

lines and diameters in a circle• Relationships involving chords of

a circle

Materials• Rulers• Scissors• BLM 12: Problem 10: Statements

for a Proof• BLM 13: Problem 10: Reasons

for Proof Statements

Chunking the Activity#1–2 #9 #14#3 #10 #15#4–6 #11 #7–8 #12–13

12 Group Presentation, Think/Pair/Share, Quickwrite, Interactive Word Wall, Vocabulary Organizer, Debriefi ng Have students brainstorm individually before comparing their responses with their group and presenting to the class. Once the two items have been debriefed, the terms tangent line, secant, and point of tangencyshould be added to the interactive word wall and students should add them to their vocabulary organizer.

3 Create Representations, Quickwrite This item is meant as a quick recognition of the theorem: If a radius is drawn to a point of tangency, the radius and tangent line are perpendicular.

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278 SpringBoard® Mathematics with Meaning™ Geometry

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278 SpringBoard® Mathematics with MeaningTM Geometry

My Notes

Chuck Goodnight dug up part of a wooden wagon wheel. An authentic western wagon has two di! erent sized wheels. " e front wheels are 42 inches in diameter while the rear wheels are 52 inches in diameter. Chuck wants to use the part of the wheel that he found to calculate the diameter of the entire wheel, so that he can determine if he has found part of a front or rear wheel. A scale drawing of Chuck’s wagon wheel part is shown below.

4. Trace the outer edge of the portion of the wheel shown onto a piece of paper.

5. Draw two chords on your arc.

6. Using a ruler, draw a perpendicular bisector to each of the two chords and extend the bisectors until they intersect.

" e perpendicular bisectors of two chords in a circle intersect at the center of the circle.

7. Determine the diameter of the circle that will be formed. Explain how you arrived at your answer.

8. " e scale factor for the drawing is 1:12. Determine which type of wheel can contain the part Chuck found. Justify your answer.

SUGGESTED LEARNING STRATEGIES: Create Representations, Use Manipulatives, Notetaking, Quickwrite

ACADEMIC VOCABULARY

An arc is part of a circle consisting of two points on the circle and the unbroken part of the circle between the two points.

Tangents and Chords ACTIVITY 4.1continued Off On a TangentOff On a Tangent

The diameter is a segment, or the length of a segment, that contains the center of a circle and two end points on the circle.

MATH TERMS

ACADEMIC VOCABULARY

A chord is a segment whose endpoints are points on a circle.

Students should follow the steps as given.



The diameter of the circle containing the given arc is 3.5 inches. Students will arrive at this answer by measuring the distance from the point of intersection of the two perpendicular bisectors created in Item 6 to a point on the circle, the radius, then doubling it.

A front wheel can contain the part that Chuck found. Solving the proportion 1 ___ 12 = 3.5 ___ x gives the diameter of the wheel as 42 inches, which is the diameter of the front wheel.

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ACTIVITY 4.1 Continued

46 Create Representations Before beginning this section, students may need to be reminded about the defi nitions of chord, arc, and perpendicularbisector. Directing students to use their vocabulary organizer will teach them that when they don’t know a defi nition they should look it up.

In Item 4, it is important that students use the outside edge of the wheel.

In Item 5, the placement of the two chords is irrelevant.

In Item 6, students should be careful when measuring to fi nd the midpoint of each chord before drawing the perpendicular bisectors.

Students should record the theorem stated after Item 6 in their notebooks, as this theorem will be important in solving problems throughout Unit 4.

78 Use Manipulatives, Note Taking, Quickwrite, Debriefi ng To determine the diameter of the scale drawing of the wheel, students must measure the radius of the arc (the distance from the point of intersection of the perpendicular bisectors to the arc). To determine the diameter of the actual wheel, students can write and solve a proportion that uses the measurements made in Item 7 and the scale factor.


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My Notes

ACTIVITY 4.1continued

Tangents and ChordsOff On a TangentOff On a Tangent

SUGGESTED LEARNING STRATEGIES: Look for a Pattern, Use Manipulatives, Quickwrite

9. In the circle below:

• Draw a diameter. • Draw a chord that is perpendicular to the diameter.

a. Use a ruler to take measurements in this ! gure. What do you notice?

b. Compare your answer with your neighbor’s answer. What conjecture can you make based on your investigations of a diameter perpendicular to a chord?


Using their measurements, students should recognize that the diameter bisects the chord.

When a diameter is perpendicular to a chord, the diameter bisects the chord.

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9 Look for a Pattern, Use Manipulatives, Quickwrite Students will use a ruler to create an illustration of and investigate the following theorem: If a diameter is perpendicular to a chord, then it bisects the chord. Students should be encouraged to compose their own conjectures before comparing with a neighbor and with the group.

Suggested Assignment

CHECK YOUR UNDERSTANDING p. 284, #1–3, 7

UNIT 4 PRACTICEp. 343, #1–2


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My Notes


10. For the theorem below, the statements for the proof have been scrambled. Your teacher will give you a sheet that lists these statements. Cut out each of the statements and rearrange them in logical order.

! eorem: In a circle, two congruent chords are equidistant from the center of the circle.

Given: __

AB ! __

CD ; __

RX " __

CD

__ RY "

__ AB

Prove: __

RY ! __

RX

Draw radii __

RB and __

RD .

__

RY ! __

RX

__

AB ! __

CD ; __

RX " __

CD ; __

RY " __

AB

AB = CD

#DXR and #BYR are right triangles.

__

RB ! __

RD

1 __ 2 AB = 1 __ 2 CD

__

BY ! __

DX

$DXR and $BYR are right angles.

BY = DX

BY = 1 __ 2 AB; DX = 1 __ 2 CD

#DXR ! #BYR

SUGGESTED LEARNING STRATEGIES: Create Representations, Notetaking, Self/Peer Revision

If two segments are the same distance from a point, they are equidistant from it.

MATH TERMS

BA Y

R

D

X

C

Answers may vary. See sample proof below.

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0 Debriefi ng, Create Representations, Note Taking, Self/Peer Revision After conducting a short class discussion about the term equidistant, distribute BLM 12 and have students cut out the statements of the proof. Next, have students work in pairs to correctly arrange these. Then have each pair compare their work with that of other students by rotating to a new pair’s work and examining that work without changing anything. After a few rotations, students should return to their own statements and make any changes that they think are necessary. Students should be aware that there is more than one correct arrangement of the statements in this proof.

It is very important to do the debriefi ng before having students move on to Item 11, in which students will arrange the reasons for the proof.

Statements 1. Draw radii

___RB and

___RD .

2. ___RB !

___RD

3. ___AB !

___CD ;

___RX "

___CD ;

___RY "

___AB

4. AB = CD

5. 1__2 AB = 1__

2 CD

6. BY = 1__2 AB; DX = 1__

2 CD

7. BY = DX

8. ___BY !

___DX

9. $DXR and $BYR are right angles.

10. #DXR and #BYR are right triangles

11. #DXR ! #BYR

12. ___RY !

___RX


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My Notes



11. ! e reasons for the proof in Item 10 are scrambled below. Your teacher will give you a sheet that lists these reasons. Cut out each of the reasons and rearrange them so they match the appropriate statement in your proof.

! rough any two points there is exactly one line.

De" nition of right triangle

De" nition of congruent segments

De" nition of congruent segments

Multiplication Property

C.P.C.T.C.

HL ! eorem

Given

De" nition of perpendicular lines

All radii of a circle are congruent.

A diameter perpendicular to a chord bisects the chord.

Substitution Property

SUGGESTED LEARNING STRATEGIES: Create Representations, Self/Peer Revision

Answers may vary. See sample proof below.

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Off On a Tangent

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a Debriefi ng, Create Representations, Self/Peer Revision Distribute BLM 13. Using the same strategy as with the statements in Item 10, have students cut out the reasons for the proof, work in pairs to arrange those reasons to correspond to their statements, and then rotate to compare their work with that of other pairs of students. After students have revised their proofs and you have debriefed the class or checked the work of each pair, have students copy their proof into their notebooks, or attach the statements and corresponding reasons to a page in their notebooks.

Reasons 1. Through any two points there

is exactly one line.

2. All radii of a circle are congruent.

3. Given

4. Defi nition of congruent segments

5. Multiplication Property

6. A diameter perpendicular to a chord bisects the chord.

7. Substitution Property

8. Defi nition of congruent segments

9. Defi nition of perpendicular lines

10. Defi nition of right triangle

11. HL Theorem

12. C.P.C.T.C.


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My Notes


SUGGESTED LEARNING STRATEGIES: Quickwrite, Think/Pair/Share

12. Given __

EF ! __

AB , explain how you know that __

EF and __

AB are not equidistant from the center, R.

13. Michael said that if two chords are the same length but are in di! erent circles that are not necessarily concentric circles, then they will not be the same distance from the center of the circle. Is he correct? If he is, give a justi" cation. If not, give a counterexample.

R

E F

A B

Answers may vary. Sample answer: ——

EF is a chord of the inner circle with center R, while

—— AB is outside the inner circle. So, the distance

from ——

AB to R is greater than the distance from ——

EF to R.

Michael is wrong. If the two circles have the same radius, then the two congruent chords will be the same distance from the centers of the circles.

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bc Quickwrite, Debriefi ng, Think/Pair/Share The purpose of these questions is to extend the theorem in Item 10 to include congruent circles. Students should be given the opportunity to think about Items 12 and 13 before discussing a solution with a partner. Allowing students to compare answers with classmates will honor multiple correct solutions to these statements.


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My Notes



SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Create Representations, Self/Peer Revision

! eorem: ! e tangent segments to a circle from a point outside the circle are congruent.

14. Use the theorem above to write the prove statement for the diagram below. ! en, prove the theorem.

Given: __

BD and __

DC are tangent to circle A.

Prove:

15. In the diagram, RT = 12 cm, RH = 5 cm, and MT = 21 cm. Determine the length of

__ RM . Explain how you arrived at your answer.

A tangent segment to a circle is part of a tangent line with one endpoint outside the circle and the second endpoint at a point of tangency to the circle.

MATH TERMS

A

C

B

D

O

D

M

AH

R

T

__

DB ! __

DC

Statements Reasons

1. __

BD and __

DC are tangent to circle A

1. Given

2. __

BA " __

BD , __

CA " __

CD 2. Tangent lines are perpendicular to the radius drawn to the point of tangency.

3. #DBA and #DCA are right angles.

3. Defi nition of perpendicular

4. $DBA and $DCA are right triangles.

4. Defi nition of right triangle

5. __

BA ! __

AC 5. All radii of a circle are congruent.

6. __

DA ! __

DA 6. Refl exive Property

7. $DBA ! $DCA 7. HL Theorem

8. __

DB ! __

DC 8. C.P.C.T.C.

See answer and explanation given at the right.

Answers may vary. Students may choose which type of proof to write. A sample two-column proof is shown.

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de Think/Pair/Share, Create Representations, Self/Peer Revision The proof of this theorem involves congruent triangles. Encourage students to write a plan before beginning the proof. Although the answer key shows a two-column proof, the proof can be written in paragraph, fl ow chart or two-column format.

It is important that students understand the implications of the theorem in Item 14 and how it can be applied in problem situations. Item 15 requires students to apply this theorem.

15. RM = 19 cm. Since two tangent segments to a circle from a point outside the circle are congruent, RD = RH =5 cm. Similarly, TA = HT, which means that TA =(RT - RH) = 12 cm - 5 cm =7 cm. AM = MT - AT =21 cm - 7 = 14 cm. Since AM and MD are both tangent to the circle from point M, AM = MD = 14 cm. RM =MD + RD = 14 cm + 5 cm =19 cm.


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Write your answers on notebook paper. Show your work.

CHECK YOUR UNDERSTANDING

Write your answers on notebook paper. Show your work.

1. Draw a circle. Draw a line that is neither a secant line nor a tangent line to the circle.

2. In the diagram, ___

› TP is tangent to circle D.

Determine m!DTP.

D

T

P

120°

3. In the diagram, ___

› CA is tangent to circle P, the

radius of circle P is 8 cm and BC = 9 cm. AC = ? .

P

B

CA

4. In the diagram, __

NM and __

QN are tangent to circle P, the radius of circle P is 5 cm, and MN = 12 cm. QN = ? .

P

Q

M

N

5. In the diagram below, __

AH , __

AD , and __

DH are each tangent to circle Q. AT = 9, AH = 13, and AD = 15. HD = ? .

Q

MD

HT

A

6. Suppose a chord of circle is 5 inches from the center and is 24 inches long. Find the length of the radius of the circle.

7. MATHEMATICAL R E F L E C T I O N

Explain how to prove the following conjecture: If a

diameter is perpendicular to a chord, then the diameter bisects the chord.

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Suggested Assignment

CHECK YOUR UNDERSTANDING p. 284, #4–6

UNIT 4 PRACTICEp. 343, #3–4

1. Answers may vary. A correct answer will consist of a circle and a line that does not intersect the circle.

2. 30°

3. 15 cm

4. 12 cm

5. 10

6. 13 inches

7. Answers may vary. Sample answer: Draw the radii of the circle to the endpoints of the chord. Because all radii in a circle are congruent, the triangle formed is isosceles. In an isosceles triangle an altitude to the base is also a median; therefore, the diameter perpendicular to the chord must also bisect the chord.


tangents and chords activity 4.1 off on a tangent...

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