NYS COMMON CORE MATHEMATICS CURRICULUM M5 Lesson 2
GEOMETRY
Lesson 2: Circles, Chords, Diameters, and Their Relationships
21
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Lesson 2: Circles, Chords, Diameters, and Their
Relationships
Student Outcomes
Students 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. Students should be made aware that figures are not
drawn to scale.
Classwork
Opening Exercise (4 minutes)
Opening Exercise
Construct the perpendicular bisector of ๐จ๐ฉฬ ฬ ฬ ฬ below (as you did in Module 1).
Draw another line that bisects ๐จ๐ฉฬ ฬ ฬ ฬ but is not perpendicular to it.
List one similarity and one difference between the two bisectors.
Answers will vary. Both bisectors divide the segment into two shorter segments of equal length.
All points on the perpendicular bisector are equidistant from points ๐จ and ๐ฉ. Points on the other
bisector are not equidistant from points ๐จ and ๐ฉ. The perpendicular bisector meets ๐จ๐ฉฬ ฬ ฬ ฬ at right
angles. The other bisector meets at angles that are not congruent.
Recall for students the definition of equidistant.
EQUIDISTANT: A point ๐ด is said to be equidistant from two different points ๐ต and ๐ถ if ๐ด๐ต = ๐ด๐ถ.
Points ๐ต and ๐ถ 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, students define the distance from the center of a circle to a
chord. This definition allows them 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 ๐ด and ๐ต.
Draw circle ๐ด with center
๐ด and radius ๐ด๐ตฬ ฬ ฬ ฬ .
Draw circle ๐ต with center
๐ต and radius ๐ต๐ดฬ ฬ ฬ ฬ .
Label the points of
intersection as ๐ถ and ๐ท.
Draw ๐ถ๐ท โก .
NYS COMMON CORE MATHEMATICS CURRICULUM M5 Lesson 2
GEOMETRY
Lesson 2: Circles, Chords, Diameters, and Their Relationships
22
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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 ๐.
Draw a chord, and label it ๐ด๐ตฬ ฬ ฬ ฬ .
Construct the perpendicular bisector of ๐ด๐ตฬ ฬ ฬ ฬ .
What do you notice about the perpendicular bisector of ๐ด๐ตฬ ฬ ฬ ฬ ?
It passes through point ๐, the center of the circle.
Draw another chord, and label it ๐ถ๐ทฬ ฬ ฬ ฬ .
Construct the perpendicular bisector of ๐ถ๐ทฬ ฬ ฬ ฬ .
What do you notice about the perpendicular bisector of ๐ถ๐ทฬ ฬ ฬ ฬ ?
It passes through point ๐, 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.
MP.3 &
MP.7
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.
๐ถ is the center of the circle
below.
๐ถ
NYS COMMON CORE MATHEMATICS CURRICULUM M5 Lesson 2
GEOMETRY
Lesson 2: Circles, Chords, Diameters, and Their Relationships
23
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Exercises (20 minutes)
Assign one proof to each group, and then jigsaw, share, and gallery walk as students present their work.
Exercises
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 ๐ช with diameter ๐ซ๐ฌฬ ฬ ฬ ฬ , chord ๐จ๐ฉฬ ฬ ฬ ฬ , and ๐จ๐ญ = ๐ฉ๐ญ
Prove: ๐ซ๐ฌฬ ฬ ฬ ฬ โฅ ๐จ๐ฉฬ ฬ ฬ ฬ
๐จ๐ญ = ๐ฉ๐ญ Given
๐ญ๐ช = ๐ญ๐ช Reflexive property
๐จ๐ช = ๐ฉ๐ช Radii of the same circle are equal in measure.
โณ ๐จ๐ญ๐ช โ โณ ๐ฉ๐ญ๐ช SSS
๐โ ๐จ๐ญ๐ช = ๐โ ๐ฉ๐ญ๐ช Corresponding angles of congruent triangles are equal in measure.
โ ๐จ๐ญ๐ช and โ ๐ฉ๐ญ๐ช are right angles Equal angles that form a linear pair each measure ๐๐ยฐ.
๐ซ๐ฌฬ ฬ ฬ ฬ โฅ ๐จ๐ฉฬ ฬ ฬ ฬ Definition of perpendicular lines
OR
๐จ๐ญ = ๐ฉ๐ญ Given
๐จ๐ช = ๐ฉ๐ช Radii of the same circle are equal in measure.
๐โ ๐ญ๐จ๐ช = ๐โ ๐ญ๐ฉ๐ช Base angles of an isosceles triangle are equal in measure.
โณ ๐จ๐ญ๐ช โ โณ ๐ฉ๐ญ๐ช SAS
๐โ ๐จ๐ญ๐ช = ๐โ ๐ฉ๐ญ๐ช Corresponding angles of congruent triangles are equal in measure.
โ ๐จ๐ญ๐ช and โ ๐ฉ๐ญ๐ช are right angles Equal angles that form a linear pair each measure ๐๐ยฐ.
๐ซ๐ฌฬ ฬ ฬ ฬ โฅ ๐จ๐ฉฬ ฬ ฬ ฬ Definition of perpendicular lines
NYS COMMON CORE MATHEMATICS CURRICULUM M5 Lesson 2
GEOMETRY
Lesson 2: Circles, Chords, Diameters, and Their Relationships
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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.
Given: Circle ๐ช with diameter ๐ซ๐ฌฬ ฬ ฬ ฬ , chord ๐จ๐ฉฬ ฬ ฬ ฬ , and ๐ซ๐ฌฬ ฬ ฬ ฬ โฅ ๐จ๐ฉฬ ฬ ฬ ฬ
Prove: ๐ซ๐ฌฬ ฬ ฬ ฬ bisects ๐จ๐ฉฬ ฬ ฬ ฬ
๐ซ๐ฌฬ ฬ ฬ ฬ โฅ ๐จ๐ฉฬ ฬ ฬ ฬ Given
โ ๐จ๐ญ๐ช and โ ๐ฉ๐ญ๐ช are right angles Definition of perpendicular lines
โณ ๐จ๐ญ๐ช and โณ ๐ฉ๐ญ๐ช are right triangles Definition of right triangle
โ ๐จ๐ญ๐ช โ โ ๐ฉ๐ญ๐ช All right angles are congruent.
๐ญ๐ช = ๐ญ๐ช Reflexive property
๐จ๐ช = ๐ฉ๐ช Radii of the same circle are equal in measure.
โณ ๐จ๐ญ๐ช โ โณ ๐ฉ๐ญ๐ช HL
๐จ๐ญ = ๐ฉ๐ญ Corresponding sides of congruent triangles are equal in length.
๐ซ๐ฌฬ ฬ ฬ ฬ bisects ๐จ๐ฉฬ ฬ ฬ ฬ Definition of segment bisector
OR
๐ซ๐ฌฬ ฬ ฬ ฬ โฅ ๐จ๐ฉฬ ฬ ฬ ฬ Given
โ ๐จ๐ญ๐ช and โ ๐ฉ๐ญ๐ช are right angles Definition of perpendicular lines
โ ๐จ๐ญ๐ช โ โ ๐ฉ๐ญ๐ช All right angles are congruent.
๐จ๐ช = ๐ฉ๐ช Radii of the same circle are equal in measure.
๐โ ๐ญ๐จ๐ช = ๐โ ๐ญ๐ฉ๐ช Base angles of an isosceles triangle are congruent.
๐โ ๐จ๐ช๐ญ = ๐โ ๐ฉ๐ช๐ญ Two angles of triangle are equal in measure, so third angles are equal.
โณ ๐จ๐ญ๐ช โ โณ ๐ฉ๐ญ๐ช ASA
๐จ๐ญ = ๐ฉ๐ญ Corresponding sides of congruent triangles are equal in length.
๐ซ๐ฌฬ ฬ ฬ ฬ bisects ๐จ๐ฉฬ ฬ ฬ ฬ Definition of segment bisector
NYS COMMON CORE MATHEMATICS CURRICULUM M5 Lesson 2
GEOMETRY
Lesson 2: Circles, Chords, Diameters, and Their Relationships
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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,
the segment also bisects the chord, as proved in Exercise 2.
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 ๐ถ with chords ๐จ๐ฉฬ ฬ ฬ ฬ and ๐ช๐ซฬ ฬ ฬ ฬ ; ๐จ๐ฉ = ๐ช๐ซ; ๐ญ is the midpoint of ๐จ๐ฉฬ ฬ ฬ ฬ and ๐ฌ is the midpoint of ๐ช๐ซฬ ฬ ฬ ฬ .
Prove: ๐ถ๐ญ = ๐ถ๐ฌ
๐จ๐ฉ = ๐ช๐ซ Given
๐ถ๐ญฬ ฬ ฬ ฬ and ๐ถ๐ฌฬ ฬ ฬ ฬ are portions of diameters Definition of diameter
๐ถ๐ญฬ ฬ ฬ ฬ โฅ ๐จ๐ฉฬ ฬ ฬ ฬ ; ๐ถ๐ฌฬ ฬ ฬ ฬ โฅ ๐ช๐ซฬ ฬ ฬ ฬ If a diameter of a circle bisects a chord, then the diameter must be
perpendicular to the chord.
โ ๐จ๐ญ๐ถ and โ ๐ซ๐ฌ๐ถ are right angles Definition of perpendicular lines
โณ ๐จ๐ญ๐ถ and โณ ๐ซ๐ฌ๐ถ are right triangles Definition of right triangle
๐ฌ and ๐ญ are midpoints of ๐ช๐ซฬ ฬ ฬ ฬ and ๐จ๐ฉฬ ฬ ฬ ฬ Given
๐จ๐ญ = ๐ซ๐ฌ ๐จ๐ฉ = ๐ช๐ซ and ๐ญ and ๐ฌ are midpoints of ๐จ๐ฉฬ ฬ ฬ ฬ and ๐ช๐ซฬ ฬ ฬ ฬ .
๐จ๐ถ = ๐ซ๐ถ All radii of a circle are equal in measure.
โณ ๐จ๐ญ๐ถ โ โณ ๐ซ๐ฌ๐ถ HL
๐ถ๐ฌ = ๐ถ๐ญ 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
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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 ๐ถ with chords ๐จ๐ฉฬ ฬ ฬ ฬ and ๐ช๐ซฬ ฬ ฬ ฬ ; ๐ถ๐ญ = ๐ถ๐ฌ; ๐ญ is the midpoint of ๐จ๐ฉฬ ฬ ฬ ฬ and ๐ฌ is the midpoint of ๐ช๐ซฬ ฬ ฬ ฬ .
Prove: ๐จ๐ฉ = ๐ช๐ซ
๐ถ๐ญ = ๐ถ๐ฌ Given
๐ถ๐ญฬ ฬ ฬ ฬ and ๐ถ๐ฌฬ ฬ ฬ ฬ are portions of diameters Definition of diameter
๐ถ๐ญฬ ฬ ฬ ฬ โฅ ๐จ๐ฉฬ ฬ ฬ ฬ ; ๐ถ๐ฌฬ ฬ ฬ ฬ โฅ ๐ช๐ซฬ ฬ ฬ ฬ If a diameter of a circle bisects a chord, then it must be perpendicular
to the chord.
โ ๐จ๐ญ๐ถ and โ ๐ซ๐ฌ๐ถ are right angles Definition of perpendicular lines
โณ ๐จ๐ญ๐ถ and โณ ๐ซ๐ฌ๐ถ are right triangles Definition of right triangle
๐จ๐ถ = ๐ซ๐ถ All radii of a circle are equal in measure.
โณ ๐จ๐ญ๐ถ โ โณ ๐ซ๐ฌ๐ถ HL
๐จ๐ญ = ๐ซ๐ฌ Corresponding sides of congruent triangles are equal in length.
๐ญ is the midpoint of ๐จ๐ฉฬ ฬ ฬ ฬ , and ๐ฌ is the
midpoint of ๐ช๐ซฬ ฬ ฬ ฬ .
Given
๐จ๐ฉ = ๐ช๐ซ ๐จ๐ญ = ๐ซ๐ฌ and ๐ญ and ๐ฌ are midpoints of ๐จ๐ฉฬ ฬ ฬ ฬ and ๐ช๐ซฬ ฬ ฬ ฬ .
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 (๐จ๐ฉฬ ฬ ฬ ฬ and ๐ช๐ซฬ ฬ ฬ ฬ ) are congruent. Since all radii of a circle are congruent, ๐จ๐ถฬ ฬ ฬ ฬ โ ๐ฉ๐ถฬ ฬ ฬ ฬ ฬ โ
๐ช๐ถฬ ฬ ฬ ฬ โ ๐ซ๐ถฬ ฬ ฬ ฬ ฬ . Therefore, โณ ๐จ๐ฉ๐ถ โ โณ ๐ซ๐ช๐ถ by SSS. โ ๐จ๐ถ๐ฉ โ โ ๐ซ๐ถ๐ช 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
27
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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, we now are given that ๐โ ๐จ๐ถ๐ฉ = ๐โ ๐ช๐ถ๐ซ. Since all radii of a circle are
congruent, ๐จ๐ถฬ ฬ ฬ ฬ โ ๐ฉ๐ถฬ ฬ ฬ ฬ ฬ โ ๐ช๐ถฬ ฬ ฬ ฬ โ ๐ซ๐ถฬ ฬ ฬ ฬ ฬ . Therefore, โณ ๐จ๐ฉ๐ถ โ โณ ๐ซ๐ช๐ถ by SAS. ๐จ๐ฉฬ ฬ ฬ ฬ โ ๐ซ๐ชฬ ฬ ฬ ฬ 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 equidistant
from center
If two chords are congruent, then the
center of a circle is equidistant from
the two chords.
If the center of a circle 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
28
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Lesson Summary
Theorems about chords and diameters in a circle and their converses:
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.
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 define central angles equal in measure.
If two chords define central angles equal in measure, then they are congruent.
Relevant Vocabulary
EQUIDISTANT: A point ๐จ is said to be equidistant from two different points ๐ฉ and ๐ช if ๐จ๐ฉ = ๐จ๐ช.
Exit Ticket (5 minutes)
NYS COMMON CORE MATHEMATICS CURRICULUM M5 Lesson 2
GEOMETRY
Lesson 2: Circles, Chords, Diameters, and Their Relationships
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Name Date
Lesson 2: Circles, Chords, Diameters, and Their Relationships
Exit Ticket
1. Given circle ๐ด shown, ๐ด๐น = ๐ด๐บ and ๐ต๐ถ = 22. Find ๐ท๐ธ.
2. In the figure, circle ๐ has a radius of 10. ๐ด๐ตฬ ฬ ฬ ฬ โฅ ๐ท๐ธฬ ฬ ฬ ฬ .
a. If ๐ด๐ต = 8, what is the length of ๐ด๐ถฬ ฬ ฬ ฬ ?
b. If ๐ท๐ถ = 2, what is the length of ๐ด๐ตฬ ฬ ฬ ฬ ?
NYS COMMON CORE MATHEMATICS CURRICULUM M5 Lesson 2
GEOMETRY
Lesson 2: Circles, Chords, Diameters, and Their Relationships
30
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Exit Ticket Sample Solutions
1. Given circle ๐จ shown, ๐จ๐ญ = ๐จ๐ฎ and ๐ฉ๐ช = ๐๐. Find ๐ซ๐ฌ.
๐๐
2. In the figure, circle ๐ท has a radius of ๐๐. ๐จ๐ฉฬ ฬ ฬ ฬ โฅ ๐ซ๐ฌฬ ฬ ฬ ฬ .
a. If ๐จ๐ฉ = ๐, what is the length of ๐จ๐ชฬ ฬ ฬ ฬ ?
๐
b. If ๐ซ๐ช = ๐, what is the length of ๐จ๐ฉฬ ฬ ฬ ฬ ?
๐๐
Problem Set Sample Solutions
Students should be made aware that figures in the Problem Set are not drawn to scale.
1. In this drawing, ๐จ๐ฉ = ๐๐, ๐ถ๐ด = ๐๐, and ๐ถ๐ต = ๐๐. What is ๐ช๐ต?
โ๐๐๐ โ ๐๐. ๐๐
2. In the figure to the right, ๐จ๐ชฬ ฬ ฬ ฬ โฅ ๐ฉ๐ฎฬ ฬ ฬ ฬ , ๐ซ๐ญฬ ฬ ฬ ฬ โฅ ๐ฌ๐ฎฬ ฬ ฬ ฬ , and ๐ฌ๐ญ = ๐๐. Find ๐จ๐ช.
๐๐
NYS COMMON CORE MATHEMATICS CURRICULUM M5 Lesson 2
GEOMETRY
Lesson 2: Circles, Chords, Diameters, and Their Relationships
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3. In the figure, ๐จ๐ช = ๐๐, and ๐ซ๐ฎ = ๐๐. Find ๐ฌ๐ฎ. Explain your work.
๐, โณ ๐จ๐ฉ๐ฎ is a right triangle with ๐ก๐ฒ๐ฉ๐จ๐ญ๐๐ง๐ฎ๐ฌ๐ = ๐ซ๐๐๐ข๐ฎ๐ฌ = ๐๐ and
๐จ๐ฉ = ๐๐, so ๐ฉ๐ฎ = ๐ by Pythagorean theorem. ๐ฉ๐ฎ = ๐ฎ๐ฌ = ๐.
4. In the figure, ๐จ๐ฉ = ๐๐, and ๐จ๐ช = ๐๐. Find ๐ซ๐ฌ.
๐
5. In the figure, ๐ช๐ญ = ๐, and the two concentric circles have radii of ๐๐ and ๐๐. Find ๐ซ๐ฌ.
๐
6. In the figure, the two circles have equal radii and intersect at points ๐ฉ and ๐ซ. ๐จ and ๐ช are centers of the circles.
๐จ๐ช = ๐, and the radius of each circle is ๐. ๐ฉ๐ซฬ ฬ ฬ ฬ ฬ โฅ ๐จ๐ชฬ ฬ ฬ ฬ . Find ๐ฉ๐ซ. Explain your work.
๐
๐ฉ๐จ = ๐ฉ๐ช = ๐ (radii)
๐จ๐ฎ = ๐ฎ๐ช = ๐
๐ฉ๐ฎ = ๐ (Pythagorean theorem)
๐ฉ๐ซ = ๐
NYS COMMON CORE MATHEMATICS CURRICULUM M5 Lesson 2
GEOMETRY
Lesson 2: Circles, Chords, Diameters, and Their Relationships
32
This work is derived from Eureka Math โข and licensed by Great Minds. ยฉ2015 Great Minds. eureka-math.org This file derived from GEO-M5-TE-1.3.0-10.2015
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
7. In the figure, the two concentric circles have radii of ๐ and ๐๐. Chord ๐ฉ๐ญฬ ฬ ฬ ฬ of the larger circle intersects the smaller
circle at ๐ช and ๐ฌ. ๐ช๐ฌ = ๐. ๐จ๐ซฬ ฬ ฬ ฬ โฅ ๐ฉ๐ญฬ ฬ ฬ ฬ .
a. Find ๐จ๐ซ.
๐โ๐
b. Find ๐ฉ๐ญ.
๐โ๐๐
8. In the figure, ๐จ is the center of the circle, and ๐ช๐ฉ = ๐ช๐ซ. Prove that ๐จ๐ชฬ ฬ ฬ ฬ bisects โ ๐ฉ๐ช๐ซ.
Let ๐จ๐ฌฬ ฬ ฬ ฬ and ๐จ๐ญฬ ฬ ฬ ฬ be perpendiculars from ๐จ to ๐ช๐ฉฬ ฬ ฬ ฬ and ๐ช๐ซฬ ฬ ฬ ฬ , respectively.
๐ช๐ฉ = ๐ช๐ซ Given
๐จ๐ฌ = ๐จ๐ญ If two chords are congruent, then the center
is equidistant from the two chords.
๐จ๐ช = ๐จ๐ช Reflexive property
๐โ ๐ช๐ฌ๐จ = ๐โ ๐ช๐ญ๐จ = ๐๐ยฐ Definition of perpendicular
โณ ๐ช๐ฌ๐จ โ โณ ๐ช๐ญ๐จ HL
๐โ ๐ฌ๐ช๐จ = ๐โ ๐ญ๐ช๐จ Corresponding angles of congruent triangles
are equal in measure.
๐จ๐ชฬ ฬ ฬ ฬ bisects โ ๐ฉ๐ช๐ซ 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 (๐จ๐ฉฬ ฬ ฬ ฬ and ๐ช๐ซฬ ฬ ฬ ฬ ) are congruent. Therefore, a rigid motion exists that carries
๐จ๐ฉฬ ฬ ฬ ฬ to ๐ช๐ซฬ ฬ ฬ ฬ . The same rotation that carries ๐จ๐ฉฬ ฬ ฬ ฬ to ๐ช๐ซฬ ฬ ฬ ฬ also carries ๐จ๐ถฬ ฬ ฬ ฬ to ๐ช๐ถฬ ฬ ฬ ฬ and ๐ฉ๐ถฬ ฬ ฬ ฬ ฬ to ๐ซ๐ถฬ ฬ ฬ ฬ ฬ . The angle of
rotation is the measure of โ ๐จ๐ถ๐ช, 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 ๐จ๐ถ๐ฉ โ โ ๐ช๐ถ๐ซ. Therefore, a rigid motion (a rotation) carries
โ ๐จ๐ถ๐ฉ to โ ๐ช๐ถ๐ซ. This same rotation carries ๐จ๐ถฬ ฬ ฬ ฬ to ๐ช๐ถฬ ฬ ฬ ฬ and ๐ฉ๐ถฬ ฬ ฬ ฬ ฬ to ๐ซ๐ถฬ ฬ ฬ ฬ ฬ . The angle of rotation is the measure
of โ ๐จ๐ถ๐ช, and the rotation is clockwise.
NYS COMMON CORE MATHEMATICS CURRICULUM M5 Lesson 2
GEOMETRY
Lesson 2: Circles, Chords, Diameters, and Their Relationships
33
This work is derived from Eureka Math โข and licensed by Great Minds. ยฉ2015 Great Minds. eureka-math.org This file derived from GEO-M5-TE-1.3.0-10.2015
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