Lesson 2: Circles, Chords, Diameters, and Their โฆ a circle of any radius, and identify the center...
Transcript of Lesson 2: Circles, Chords, Diameters, and Their โฆ a circle of any radius, and identify the center...
NYS COMMON CORE MATHEMATICS CURRICULUM M5 Lesson 2 GEOMETRY
Lesson 2: Circles, Chords, Diameters, and Their Relationships Date: 10/22/14
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ยฉ 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 2: Circles, Chords, Diameters, and Their
Relationships
Student Outcomes
Identify the relationships between the diameters of a circle and other chords of the circle.
Lesson Notes Students are asked to construct the perpendicular bisector of a line segment and draw conclusions about points on that bisector and the endpoints of the segment. They relate the construction to the theorem stating that any perpendicular bisector of a chord must pass through the center of the circle.
Classwork
Opening Exercise (4 minutes)
Opening Exercise
Construct the perpendicular bisector of line segment ๐จ๐จ๐จ๐จ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ below (as you did in Module 1).
Draw another line that bisects ๐จ๐จ๐จ๐จ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ but is not perpendicular to it.
List one similarity and one difference between the two bisectors.
Answers will vary. All points on the perpendicular bisector are equidistant from points ๐จ๐จ and ๐จ๐จ46T. Points on the other bisector are not equidistant from points A and B. The perpendicular bisector meets ๐จ๐จ๐จ๐จ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ at right angles. The other bisector meets at angles that are not congruent.
You may wish to recall for students the definition of equidistant:
EQUIDISTANT:. A point ๐ด๐ด is said to be equidistant from two different points ๐ต๐ต and ๐ถ๐ถ if ๐ด๐ด๐ต๐ต = ๐ด๐ด๐ถ๐ถ.
Points ๐ต๐ต and ๐ถ๐ถ can be replaced in the definition above with other figures (lines, etc.) as long as the distance to those figures is given meaning first. In this lesson, we will define the distance from the center of a circle to a chord. This definition will allow us to talk about the center of a circle as being equidistant from two chords.
Scaffolding: Post a diagram and display the steps to create a perpendicular bisector used in Lesson 4 of Module 1. Label the endpoints of the
segment ๐ด๐ด and ๐ต๐ต. Draw circle ๐ด๐ด with center
๐ด๐ด and radius ๐ด๐ด๐ต๐ต๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ. Draw circle ๐ต๐ต with center
๐ต๐ต and radius ๐ต๐ต๐ด๐ด๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ. Label the points of
intersection as ๐ถ๐ถ and ๐ท๐ท.
Draw ๐ถ๐ถ๐ท๐ท๏ฟฝโ๏ฟฝ๏ฟฝ๏ฟฝโ .
NYS COMMON CORE MATHEMATICS CURRICULUM M5 Lesson 2 GEOMETRY
Lesson 2: Circles, Chords, Diameters, and Their Relationships Date: 10/22/14
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Discussion (12 minutes)
Ask students independently or in groups to each draw chords and describe what they notice. Answers will vary depending on what each student drew.
Lead students to relate the perpendicular bisector of a line segment to the points on a circle, guiding them toward seeing the relationship between the perpendicular bisector of a chord and the center of a circle.
Construct a circle of any radius, and identify the center as point ๐๐.
Draw a chord, and label it ๐ด๐ด๐ต๐ต๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ.
Construct the perpendicular bisector of ๐ด๐ด๐ต๐ต๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ.
What do you notice about the perpendicular bisector of ๐ด๐ด๐ต๐ต๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ? It passes through point P, the center of the circle.
Draw another chord and label it ๐ถ๐ถ๐ท๐ท๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ.
Construct the perpendicular bisector of ๐ถ๐ถ๐ท๐ท๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ.
What do you notice about the perpendicular bisector of ๐ถ๐ถ๐ท๐ท๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ? It passes through point ๐๐, the center of the circle.
What can you say about the points on a circle in relation to the center of the circle?
The center of the circle is equidistant from any two points on the circle.
Look at the circles, chords, and perpendicular bisectors created by your neighbors. What statement can you make about the perpendicular bisector of any chord of a circle? Why?
It must contain the center of the circle. The center of the circle is equidistant from the two endpoints of the chord because they lie on the circle. Therefore, the center lies on the perpendicular bisector of the chord. That is, the perpendicular bisector contains the center.
How does this relate to the definition of the perpendicular bisector of a line segment? The set of all points equidistant from two given points (endpoints of a
line segment) is precisely the set of all points on the perpendicular bisector of the line segment.
Scaffolding: Review the definition of
central angle by posting a visual guide.
A central angle of a circle is an angle whose vertex is the center of a circle.
MP.3 &
MP.7
NYS COMMON CORE MATHEMATICS CURRICULUM M5 Lesson 2 GEOMETRY
Lesson 2: Circles, Chords, Diameters, and Their Relationships Date: 10/22/14
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Exercises 1โ6 (20 minutes)
Assign one proof to each group, and then jigsaw, share, and gallery walk as students present their work.
Exercises 1โ6
1. Prove the theorem: If a diameter of a circle bisects a chord, then it must be perpendicular to the chord.
Draw a diagram similar to that shown below.
Given: Circle ๐ช๐ช with diameter ๐ซ๐ซ๐ซ๐ซ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ, chord ๐จ๐จ๐จ๐จ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ , and ๐จ๐จ๐จ๐จ = ๐จ๐จ๐จ๐จ.
Prove: ๐ซ๐ซ๐ซ๐ซ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โฅ ๐จ๐จ๐จ๐จ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ
๐จ๐จ๐จ๐จ = ๐จ๐จ๐จ๐จ Given
๐จ๐จ๐ช๐ช = ๐จ๐จ๐ช๐ช Reflexive property
๐จ๐จ๐ช๐ช = ๐จ๐จ๐ช๐ช Radii of the same circle are equal in measure
โณ ๐จ๐จ๐จ๐จ๐ช๐ช โ โณ ๐จ๐จ๐จ๐จ๐ช๐ช SSS
๐๐โ ๐จ๐จ๐จ๐จ๐ช๐ช = ๐๐โ ๐จ๐จ๐จ๐จ๐ช๐ช Corresponding angles of congruent triangles are equal in measure
โ ๐จ๐จ๐จ๐จ๐ช๐ช and โ ๐จ๐จ๐จ๐จ๐ช๐ช are right angles Equal angles that form a linear pair each measure ๐๐๐๐ยฐ
๐ซ๐ซ๐ซ๐ซ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โฅ ๐จ๐จ๐จ๐จ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ Definition of perpendicular lines
OR
๐จ๐จ๐จ๐จ = ๐จ๐จ๐จ๐จ Given
๐จ๐จ๐ช๐ช = ๐จ๐จ๐ช๐ช Radii of the same circle are equal in measure
๐๐โ ๐จ๐จ๐จ๐จ๐ช๐ช = ๐๐โ ๐จ๐จ๐จ๐จ๐ช๐ช Base angles of an isosceles are equal in measure
โณ ๐จ๐จ๐จ๐จ๐ช๐ช โ โณ๐จ๐จ๐จ๐จ๐ช๐ช SAS
๐๐โ ๐จ๐จ๐จ๐จ๐ช๐ช = ๐๐โ ๐จ๐จ๐จ๐จ๐ช๐ช Corresponding angles of congruent triangles are equal in measure
โ ๐จ๐จ๐จ๐จ๐ช๐ช and โ ๐จ๐จ๐จ๐จ๐ช๐ช are right angles Equal angles that form a linear pair each measure ๐๐๐๐ยฐ
๐ซ๐ซ๐ซ๐ซ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โฅ ๐จ๐จ๐จ๐จ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ Definition of perpendicular lines
NYS COMMON CORE MATHEMATICS CURRICULUM M5 Lesson 2 GEOMETRY
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2. Prove the theorem: If a diameter of a circle is perpendicular to a chord, then it bisects the chord.
Use a diagram similar to that in Exercise 1 above.
Given: Circle ๐ช๐ช with diameter ๐ซ๐ซ๐ซ๐ซ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ, chord ๐จ๐จ๐จ๐จ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ , and ๐ซ๐ซ๐ซ๐ซ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โฅ ๐จ๐จ๐จ๐จ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ
Prove: ๐ซ๐ซ๐ซ๐ซ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ bisects ๐จ๐จ๐จ๐จ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ
๐ซ๐ซ๐ซ๐ซ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โฅ ๐จ๐จ๐จ๐จ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ Given
โ ๐จ๐จ๐จ๐จ๐ช๐ช and โ ๐จ๐จ๐จ๐จ๐ช๐ช are right angles Definition of perpendicular lines
โณ ๐จ๐จ๐จ๐จ๐ช๐ช and โณ๐จ๐จ๐จ๐จ๐ช๐ช are right triangles Definition of right triangle
โ ๐จ๐จ๐จ๐จ๐ช๐ช โ โ ๐จ๐จ๐จ๐จ๐ช๐ช All right angles are congruent
๐จ๐จ๐ช๐ช = ๐จ๐จ๐ช๐ช Reflexive property
๐จ๐จ๐ช๐ช = ๐จ๐จ๐ช๐ช Radii of the same circle are equal in measure
โณ ๐จ๐จ๐จ๐จ๐ช๐ช โ โณ๐จ๐จ๐จ๐จ๐ช๐ช HL
๐จ๐จ๐จ๐จ = ๐จ๐จ๐จ๐จ Corresponding sides of congruent triangles are equal in length
๐ซ๐ซ๐ซ๐ซ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ bisects ๐จ๐จ๐จ๐จ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ Definition of segment bisector
OR
๐ซ๐ซ๐ซ๐ซ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โฅ ๐จ๐จ๐จ๐จ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ Given
โ ๐จ๐จ๐จ๐จ๐ช๐ช and โ ๐จ๐จ๐จ๐จ๐ช๐ช are right angles Definition of perpendicular lines
โ ๐จ๐จ๐จ๐จ๐ช๐ช โ โ ๐จ๐จ๐จ๐จ๐ช๐ช All right angles are congruent
๐จ๐จ๐ช๐ช = ๐จ๐จ๐ช๐ช Radii of the same circle are equal in measure
๐๐โ ๐จ๐จ๐จ๐จ๐ช๐ช = ๐๐โ ๐จ๐จ๐จ๐จ๐ช๐ช Base angles of an isosceles triangle are congruent
๐๐โ ๐จ๐จ๐ช๐ช๐จ๐จ = ๐๐โ ๐จ๐จ๐ช๐ช๐จ๐จ Two angles of triangle are equal in measure, so third angles are equal
โณ ๐จ๐จ๐จ๐จ๐ช๐ช โ โณ ๐จ๐จ๐จ๐จ๐ช๐ช ASA
๐จ๐จ๐จ๐จ = ๐จ๐จ๐จ๐จ Corresponding sides of congruent triangles are equal in length
๐ซ๐ซ๐ซ๐ซ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ bisects ๐จ๐จ๐จ๐จ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ Definition of segment bisector
NYS COMMON CORE MATHEMATICS CURRICULUM M5 Lesson 2 GEOMETRY
Lesson 2: Circles, Chords, Diameters, and Their Relationships Date: 10/22/14
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3. The distance from the center of a circle to a chord is defined as the length of the perpendicular segment from the center to the chord. Note that, since this perpendicular segment may be extended to create a diameter of the circle, therefore, the segment also bisects the chord, as proved in Exercise 2 above.
Prove the theorem: In a circle, if two chords are congruent, then the center is equidistant from the two chords.
Use the diagram below.
Given: Circle ๐ถ๐ถ with chords ๐จ๐จ๐จ๐จ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ and ๐ช๐ช๐ซ๐ซ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ; ๐จ๐จ๐จ๐จ = ๐ช๐ช๐ซ๐ซ; ๐จ๐จ is the midpoint of ๐จ๐จ๐จ๐จ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ and ๐ซ๐ซ is the midpoint of ๐ช๐ช๐ซ๐ซ.
Prove: ๐ถ๐ถ๐จ๐จ = ๐ถ๐ถ๐ซ๐ซ
๐จ๐จ๐จ๐จ = ๐ช๐ช๐ซ๐ซ Given
๐ถ๐ถ๐จ๐จ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โฅ ๐จ๐จ๐จ๐จ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ; ๐ถ๐ถ๐ซ๐ซ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โฅ ๐ช๐ช๐ซ๐ซ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ If a diameter of a circle bisects a chord, then the diameter must be perpendicular to the chord.
โ ๐จ๐จ๐จ๐จ๐ถ๐ถ and โ ๐ซ๐ซ๐ซ๐ซ๐ถ๐ถ are right angles Definition of perpendicular lines
โณ ๐จ๐จ๐จ๐จ๐ถ๐ถ and โณ๐ซ๐ซ๐ซ๐ซ๐ถ๐ถ are right triangles Definition of right triangle
E and ๐จ๐จ are midpoints of ๐ช๐ช๐ซ๐ซ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ and ๐จ๐จ๐จ๐จ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ Given
๐จ๐จ๐จ๐จ = ๐ซ๐ซ๐ซ๐ซ ๐จ๐จ๐จ๐จ = ๐ช๐ช๐ซ๐ซ and ๐จ๐จ and ๐ซ๐ซ are midpoints of ๐จ๐จ๐จ๐จ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ and ๐ช๐ช๐ซ๐ซ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ
๐จ๐จ๐ถ๐ถ = ๐ซ๐ซ๐ถ๐ถ All radii of a circle are equal in measure
โณ ๐จ๐จ๐จ๐จ๐ถ๐ถ โ โณ๐ซ๐ซ๐ซ๐ซ๐ถ๐ถ HL
๐ถ๐ถ๐ซ๐ซ = ๐ถ๐ถ๐จ๐จ Corresponding sides of congruent triangles are equal in length
NYS COMMON CORE MATHEMATICS CURRICULUM M5 Lesson 2 GEOMETRY
Lesson 2: Circles, Chords, Diameters, and Their Relationships Date: 10/22/14
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4. Prove the theorem: In a circle, if the center is equidistant from two chords, then the two chords are congruent.
Use the diagram below.
Given: Circle ๐ถ๐ถ with chords ๐จ๐จ๐จ๐จ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ and ๐ช๐ช๐ซ๐ซ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ; ๐ถ๐ถ๐จ๐จ = ๐ถ๐ถ๐ซ๐ซ; ๐จ๐จ is the midpoint of ๐จ๐จ๐จ๐จ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ and ๐ซ๐ซ is the midpoint of ๐ช๐ช๐ซ๐ซ
Prove: ๐จ๐จ๐จ๐จ = ๐ช๐ช๐ซ๐ซ
๐ถ๐ถ๐จ๐จ = ๐ถ๐ถ๐ซ๐ซ Given
๐ถ๐ถ๐จ๐จ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โฅ ๐จ๐จ๐จ๐จ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ; ๐ถ๐ถ๐ซ๐ซ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โฅ ๐ช๐ช๐ซ๐ซ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ If a diameter of a circle bisects a chord, then it must be perpendicular to the chord.
โ ๐จ๐จ๐จ๐จ๐ถ๐ถ and โ ๐ซ๐ซ๐ซ๐ซ๐ถ๐ถ are right angles Definition of perpendicular lines.
โณ ๐จ๐จ๐จ๐จ๐ถ๐ถ and โณ๐ซ๐ซ๐ซ๐ซ๐ถ๐ถ are right triangles Definition of right triangle.
๐จ๐จ๐ถ๐ถ = ๐ซ๐ซ๐ถ๐ถ All radii of a circle are equal in measure.
โณ ๐จ๐จ๐จ๐จ๐ถ๐ถ โ โณ๐ซ๐ซ๐ซ๐ซ๐ถ๐ถ HL
๐จ๐จ๐จ๐จ = ๐ซ๐ซ๐ซ๐ซ Corresponding sides of congruent triangles are equal in length.
๐จ๐จ is the midpoint of ๐จ๐จ๐จ๐จ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ, and ๐ซ๐ซ is the midpoint of ๐ช๐ช๐ซ๐ซ Given
๐จ๐จ๐จ๐จ = ๐ช๐ช๐ซ๐ซ ๐จ๐จ๐จ๐จ = ๐ซ๐ซ๐ซ๐ซ and ๐จ๐จ and ๐ซ๐ซ are midpoints of ๐จ๐จ๐จ๐จ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ and ๐ช๐ช๐ซ๐ซ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ.
5. A central angle defined by a chord is an angle whose vertex is the center of the circle and whose rays intersect the circle. The points at which the angleโs rays intersect the circle form the endpoints of the chord defined by the central angle. Prove the theorem: In a circle, congruent chords define central angles equal in measure.
Use the diagram below.
We are given that the two chords (๐จ๐จ๐จ๐จ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ and ๐ช๐ช๐ซ๐ซ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ) are congruent. Since all radii of a circle are congruent, ๐จ๐จ๐ถ๐ถ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โ ๐จ๐จ๐ถ๐ถ โ ๐ช๐ช๐ถ๐ถ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โ ๐ซ๐ซ๐ถ๐ถ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ. Therefore, โณ ๐จ๐จ๐จ๐จ๐ถ๐ถ โ โณ ๐ช๐ช๐ซ๐ซ๐ถ๐ถ by SSS. โ ๐จ๐จ๐ถ๐ถ๐จ๐จ โ โ ๐ช๐ช๐ถ๐ถ๐ซ๐ซ since corresponding angles of congruent triangles are equal in measure.
NYS COMMON CORE MATHEMATICS CURRICULUM M5 Lesson 2 GEOMETRY
Lesson 2: Circles, Chords, Diameters, and Their Relationships Date: 10/22/14
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6. Prove the theorem: In a circle, if two chords define central angles equal in measure, then they are congruent.
Using the diagram from Exercise 5 above, we now are given that ๐๐โ ๐จ๐จ๐ถ๐ถ๐จ๐จ = ๐๐ โ ๐ช๐ช๐ถ๐ถ๐ซ๐ซ. Since all radii of a circle are congruent, ๐จ๐จ๐ถ๐ถ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โ ๐จ๐จ๐ถ๐ถ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โ ๐ช๐ช๐ถ๐ถ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โ ๐ซ๐ซ๐ถ๐ถ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ. Therefore, โณ ๐จ๐จ๐จ๐จ๐ถ๐ถ โ โณ ๐ช๐ช๐ซ๐ซ๐ถ๐ถ by SAS. ๐จ๐จ๐จ๐จ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โ ๐ช๐ช๐ซ๐ซ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ because corresponding sides of congruent triangles are congruent
Closing (4 minutes)
Have students write all they know to be true about the diagrams below. Bring the class together, go through the Lesson Summary, having students complete the list that they started, and discuss each point.
A reproducible version of the graphic organizer shown is included at the end of the lesson.
Diagram Explanation of Diagram Theorem or Relationship
Diameter of a circle bisecting a chord
If a diameter of a circle bisects a chord, then it must be perpendicular
to the chord. If a diameter of a circle is
perpendicular to a chord, then it bisects the chord.
Two congruent chords equidistance from center
If two chords are congruent, then the center is equidistant from the two
chords. If the center is equidistant from two
chords, then the two chords are congruent.
Congruent chords
Congruent chords define central angles equal in measure.
If two chords define central angles equal in measure, then they are
congruent.
NYS COMMON CORE MATHEMATICS CURRICULUM M5 Lesson 2 GEOMETRY
Lesson 2: Circles, Chords, Diameters, and Their Relationships Date: 10/22/14
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Lesson Summary
Theorems about chords and diameters in a circle and their converses
โข If a diameter of a circle bisects a chord, then it must be perpendicular to the chord.
โข If a diameter of a circle is perpendicular to a chord, then it bisects the chord.
โข If two chords are congruent, then the center is equidistant from the two chords.
โข If the center is equidistant from two chords, then the two chords are congruent.
โข Congruent chords define central angles equal in measure.
โข If two chords define central angles equal in measure, then they are congruent.
Relevant Vocabulary
EQUIDISTANT: A point ๐จ๐จ is said to be equidistant from two different points ๐จ๐จ and ๐ช๐ช if ๐จ๐จ๐จ๐จ = ๐จ๐จ๐ช๐ช.
Exit Ticket (5 minutes)
NYS COMMON CORE MATHEMATICS CURRICULUM M5 Lesson 2 GEOMETRY
Lesson 2: Circles, Chords, Diameters, and Their Relationships Date: 10/22/14
29
ยฉ 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Name Date
Lesson 2: Circles, Chords, Diameters, and Their Relationships
Exit Ticket
Given circle ๐ด๐ด shown, ๐ด๐ด๐ด๐ด = ๐ด๐ด๐ด๐ด and ๐ต๐ต๐ถ๐ถ = 22. Find ๐ท๐ท๐ท๐ท. 1.
In the figure, circle ๐๐ has a radius of 10. ๐ด๐ด๐ต๐ต๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โฅ ๐ท๐ท๐ท๐ท๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ. 2.
a. If ๐ด๐ด๐ต๐ต = 8, what is the length of ๐ด๐ด๐ถ๐ถ?
b. If ๐ท๐ท๐ถ๐ถ = 2, what is the length of ๐ด๐ด๐ต๐ต?
NYS COMMON CORE MATHEMATICS CURRICULUM M5 Lesson 2 GEOMETRY
Lesson 2: Circles, Chords, Diameters, and Their Relationships Date: 10/22/14
30
ยฉ 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Exit Ticket Sample Solutions
1. Given circle ๐จ๐จ shown,๐จ๐จ๐จ๐จ = ๐จ๐จ๐จ๐จ and ๐จ๐จ๐ช๐ช = ๐๐๐๐. Find ๐ซ๐ซ๐ซ๐ซ. ๐๐๐๐
2. In the figure, circle ๐ท๐ท has a radius of ๐๐๐๐. ๐จ๐จ๐จ๐จ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โฅ ๐ซ๐ซ๐ซ๐ซ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ. a. If ๐จ๐จ๐จ๐จ = ๐๐, what is the length of ๐จ๐จ๐ช๐ช?
๐๐
b. If ๐ซ๐ซ๐ช๐ช = ๐๐, what is the length of ๐จ๐จ๐จ๐จ?
๐๐๐๐
Problem Set Sample Solutions
1. In this drawing, ๐จ๐จ๐จ๐จ = ๐๐๐๐, ๐ถ๐ถ๐ถ๐ถ = ๐๐๐๐, and ๐ถ๐ถ๐ถ๐ถ = ๐๐๐๐. What is ๐ช๐ช๐ถ๐ถ?
๐๐โ๐๐๐๐๐๐ (โ ๐๐๐๐.๐๐)
2. In the figure to the right, ๐จ๐จ๐ช๐ช๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โฅ ๐จ๐จ๐จ๐จ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ and ๐ซ๐ซ๐จ๐จ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โฅ ๐ซ๐ซ๐จ๐จ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ; ๐ซ๐ซ๐จ๐จ = ๐๐๐๐. Find ๐จ๐จ๐ช๐ช.
๐๐๐๐
3. In the figure, ๐จ๐จ๐ช๐ช = ๐๐๐๐, and ๐ซ๐ซ๐จ๐จ = ๐๐๐๐. Find ๐ซ๐ซ๐จ๐จ. Explain your work.
๐๐, โณ ๐จ๐จ๐จ๐จ๐จ๐จ is a right triangle with ๐ก๐ก๐ก๐ก๐ก๐ก๐ก๐ก๐ก๐ก๐ก๐ก๐ก๐ก๐ก๐ก๐ก๐ก๐ก๐ก = ๐ซ๐ซ๐ซ๐ซ๐ซ๐ซ๐ซ๐ซ๐ก๐ก๐ก๐ก = ๐๐๐๐ and ๐จ๐จ๐จ๐จ = ๐๐๐๐, so ๐จ๐จ๐จ๐จ = ๐๐ by Pythagorean theorem. ๐จ๐จ๐จ๐จ = ๐จ๐จ๐ซ๐ซ = ๐๐.
NYS COMMON CORE MATHEMATICS CURRICULUM M5 Lesson 2 GEOMETRY
Lesson 2: Circles, Chords, Diameters, and Their Relationships Date: 10/22/14
31
ยฉ 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
4. In the figure, ๐จ๐จ๐จ๐จ = ๐๐๐๐, ๐จ๐จ๐ช๐ช = ๐๐๐๐. Find ๐ซ๐ซ๐ซ๐ซ.
๐๐
5. In the figure, ๐ช๐ช๐จ๐จ = ๐๐, and the two concentric circles have radii of ๐๐๐๐ and ๐๐๐๐. Find ๐ซ๐ซ๐ซ๐ซ.
๐๐
6. In the figure, the two circles have equal radii and intersect at points ๐จ๐จ and ๐ซ๐ซ. ๐จ๐จ and ๐ช๐ช are centers of the circles. ๐จ๐จ๐ช๐ช = ๐๐, and the radius of each circle is ๐๐. ๐จ๐จ๐ซ๐ซ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โฅ ๐จ๐จ๐ช๐ช๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ. Find ๐จ๐จ๐ซ๐ซ. Explain your work.
๐๐
๐จ๐จ๐จ๐จ = ๐จ๐จ๐ช๐ช = ๐๐ (radii)
๐จ๐จ๐จ๐จ = ๐จ๐จ๐ช๐ช = ๐๐
๐จ๐จ๐จ๐จ = ๐๐ (Pythagorean theorem)
๐จ๐จ๐ซ๐ซ = ๐๐
7. In the figure, the two concentric circles have radii of ๐๐ and ๐๐๐๐. Chord ๐จ๐จ๐จ๐จ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ of the larger circle intersects the smaller circle at ๐ช๐ช and ๐ซ๐ซ. ๐ช๐ช๐ซ๐ซ = ๐๐. ๐จ๐จ๐ซ๐ซ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โฅ ๐จ๐จ๐จ๐จ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ. a. Find ๐จ๐จ๐ซ๐ซ.
๐๐โ๐๐
b. Find ๐จ๐จ๐จ๐จ.
๐๐โ๐๐๐๐
NYS COMMON CORE MATHEMATICS CURRICULUM M5 Lesson 2 GEOMETRY
Lesson 2: Circles, Chords, Diameters, and Their Relationships Date: 10/22/14
32
ยฉ 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
8. In the figure, ๐จ๐จ is the center of the circle, and ๐ช๐ช๐จ๐จ = ๐ช๐ช๐ซ๐ซ. Prove that ๐จ๐จ๐ช๐ช๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ bisects โ ๐จ๐จ๐ช๐ช๐ซ๐ซ.
Let ๐จ๐จ๐ซ๐ซ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ and ๐จ๐จ๐จ๐จ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ be perpendiculars from ๐จ๐จ to ๐ช๐ช๐จ๐จ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ and ๐ช๐ช๐ซ๐ซ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ, respectively.
๐ช๐ช๐จ๐จ = ๐ช๐ช๐ซ๐ซ Given
๐จ๐จ๐ซ๐ซ = ๐จ๐จ๐จ๐จ If two chords are congruent, then the center is equidistant from the two chords.
๐จ๐จ๐ช๐ช = ๐จ๐จ๐ช๐ช Reflexive property
โ ๐ช๐ช๐ซ๐ซ๐จ๐จ = โ ๐ช๐ช๐จ๐จ๐จ๐จ = ๐๐๐๐ยฐ A line joining the center and the midpoint of a chord is perpendicular to the chord.
โณ ๐ช๐ช๐ซ๐ซ๐จ๐จ โ โณ ๐ช๐ช๐จ๐จ๐จ๐จ HL
โ ๐ซ๐ซ๐ช๐ช๐จ๐จ = โ ๐จ๐จ๐ช๐ช๐จ๐จ Corresponding angles of congruent triangles are equal in measure.
๐จ๐จ๐ช๐ช๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ bisects โ ๐จ๐จ๐ช๐ช๐ซ๐ซ Definition of angle bisector
9. In class, we proved: Congruent chords define central angles equal in measure.
a. Give another proof of this theorem based on the properties of rotations. Use the figure from Exercise 5.
We are given that the two chords (๐จ๐จ๐จ๐จ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ and ๐ช๐ช๐ซ๐ซ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ) are congruent. Therefore, a rigid motion exists that carries ๐จ๐จ๐จ๐จ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ to ๐ช๐ช๐ซ๐ซ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ. The same rotation that carries ๐จ๐จ๐จ๐จ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ to ๐ช๐ช๐ซ๐ซ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ also carries ๐จ๐จ๐ถ๐ถ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ to ๐ช๐ช๐ถ๐ถ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ and ๐จ๐จ๐ถ๐ถ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ to ๐ซ๐ซ๐ถ๐ถ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ. The angle of rotation is the measure of โ ๐จ๐จ๐ถ๐ถ๐ช๐ช, and the rotation is clockwise.
b. Give a rotation proof of the converse: If two chords define central angles of the same measure, then they must be congruent.
Using the same diagram, we are given that โ ๐จ๐จ๐ถ๐ถ๐จ๐จ โ โ ๐ช๐ช๐ถ๐ถ๐ซ๐ซ. Therefore, a rigid motion (a rotation) carries โ ๐จ๐จ๐ถ๐ถ๐จ๐จ to โ ๐ช๐ช๐ถ๐ถ๐ซ๐ซ. This same rotation carries ๐จ๐จ๐ถ๐ถ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ to ๐ช๐ช๐ถ๐ถ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ and ๐จ๐จ๐ถ๐ถ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ to ๐ซ๐ซ๐ถ๐ถ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ. The angle of rotation is the measure of โ ๐จ๐จ๐ถ๐ถ๐ช๐ช, and the rotation is clockwise.
NYS COMMON CORE MATHEMATICS CURRICULUM M5 Lesson 2 GEOMETRY
Lesson 2: Circles, Chords, Diameters, and Their Relationships Date: 10/22/14
33
ยฉ 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Graphic Organizer on Circles
Diagram Explanation of Diagram Theorem or Relationship