MM2G3 Students will understand properties of circles.
MM2G3 d Justify measurements and relationships in circles using geometric and algebraic properties.
Apply Properties of Chords
Essential Question:
How do we use relationships of arcs and chords in a circle?
M2 Unit 3: Day 3
Lesson 6.3
Wednesday, April 19, 2023
MM2G3 Students will understand properties of circles.
MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.
Daily Homework Quiz
Describe each figure as a minor arc, major arc or a semicircle. Find the arc measure.
1. BC
ANSWER minor arc, 32o
Daily Homework Quiz
MM2G3 Students will understand properties of circles.
MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.
Daily Homework Quiz
2.
Describe each figure as a minor arc, major arc or a semicircle. Find the arc measure.
CBE
ANSWER major arc, 212o
Daily Homework Quiz
MM2G3 Students will understand properties of circles.
MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.
Daily Homework Quiz
3.
Describe each figure as a minor arc, major arc or a semicircle. Find the arc measure.
BCE
ANSWER semicircle, 180o
Daily Homework Quiz
MM2G3 Students will understand properties of circles.
MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.
4.
Describe each figure as a minor arc, major arc or a semicircle. Find the arc measure.
BCAEExplain why =~ .
ANSWER
BCAE =~
m AFE = m BFC because the angles are vertical angles, so AFE BFC.Then arcs and are arcs that have the same measure in the same circle. By definition .
=~AE BC
Daily Homework Quiz
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.ACD AC
Two diameters of P are AB and CD.If m = 50 , find m and m .
. ADo
ANSWERo
310 ; 130o
Daily Homework Quiz
MM2G3 Students will understand properties of circles.
MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.
radius1. DC
Tell whether the segment is best described as a radius,chord, or diameter of C.
Warm Ups
diameter2. BD
3. DEchord
4. AE
5. Solve 4x = 8x – 12. 6. Solve 3x + 2 = 6x – 4.
x = 3 x = 2
chord
MM2G3 Students will understand properties of circles.
MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.
Theorem 6.5 In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.
C
B
A» ¼@ @ if and only if AB BC AB BC
MM2G3 Students will understand properties of circles.
MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.
Use congruent chords to find an arc measure
In the diagram, P Q, FG JK , and mJK = 80o. Find mFG
SOLUTION
Because FG and JK are congruent chords in congruent circles, the corresponding minor arcs FG and JK are congruent.
So, mFG = mJK = 80o.
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
Because AB and BC are congruent chords in the same circle, the corresponding minor arcs AB and BC are congruent.
Use the diagram of D.
1. If mAB = 110°, find mBC
So, mBC = mAB = 110o.
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 PRACTICEUse the diagram of D.
2. If mAC = 150°, find mABBecause AB and BC are congruent chords in the same circle, the corresponding minor arcs AB and BC are congruent.
Subtract
Substitute
mAB = 105° Simplify
So, mBC = mAB
And, mBC + mAB + mAC = 360°
So, 2 mAB + mAC = 360° 2 mAB + 150° = 360°
2 mAB = 360 – 150 2 mAB = 210
mAB = 105° ANSWER
MM2G3 Students will understand properties of circles.
MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.
Theorem 6.6
If one chord is a perpendicular bisector of another chord, then the first chord is a diameter.
is a diameter of the circleJK
J
L
K
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.
Theorem 6.7
If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.
E
D
G
FDE EF¼ »@DG GF
MM2G3 Students will understand properties of circles.
MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.
Use a diameter
SOLUTION
Use the diagram of E to find the length of AC . Tell what theorem you use.
Diameter BD is perpendicular to AC . So, by Theorem 6.7, BD bisects AC , and CF = AF. Therefore, AC = 2 AF = 2(7) = 14.
MM2G3 Students will understand properties of circles.
MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.
3. CD
So 9x° = (80 – x)° So 10x° = 80° x = 8°
So mCD = 9x° = 72°
From the diagramDiameter BD is perpendicular to CE . So, by Theorem 6.7, BD bisects CE , Therefore mCD = mDE.
Find the measure of the indicated arc in the diagram.
SOLUTION
MM2G3 Students will understand properties of circles.
MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.
4. DE
mCD = mDE.
So mDE = 72°
5. CE
mCE = mDE + mCD
So mCE = 72° + 72° = 144°
Find the measure of the indicated arc in the diagram.
SOLUTION
SOLUTION
MM2G3 Students will understand properties of circles.
MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.
Theorem 6.8In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center.
F
G
E
B
A
C
D
if and only if GE = FEAB CD
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
Chords QR and ST are congruent, so by Theorem 6.8 they are equidistant from C. Therefore, CU = CV.
CU = CV
2x = 5x – 9
x = 3
So, CU = 2x = 2(3) = 6.
Use Theorem 6.8
Substitute.
Solve for x.
In the diagram of C, QR = ST = 16. Find CU.
MM2G3 Students will understand properties of circles.
MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.
Since CU = CV. Therefore Chords QR and ST are equidistant from center and from theorem 6.8 QR is congruent to ST
SOLUTION
QR = STQR = 32
Use Theorem 6.8.
Substitute.
6. QR
Suppose ST = 32, and CU = CV = 12. Find the given length.
MM2G3 Students will understand properties of circles.
MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.
Since CU is the line drawn from the center of the circle to the chord QR it will bisect the chord.
SOLUTION
QU = 16
Substitute.
7. QU
2So QU = QR1
2So QU = (32)1
Suppose ST = 32, and CU = CV = 12. Find the given length.
MM2G3 Students will understand properties of circles.
MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.
Join the points Q and C. Now QUC is right angled triangle. Use the Pythagorean Theorem to find the QC which will represent the radius of the C
SOLUTION
8. The radius of C
Suppose ST = 32, and CU = CV = 12. Find the given length.
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
Suppose ST = 32, and CU = CV = 12. Find the given length.8. The radius of C
So QC2 = 162 + 122
So QC2 = 256 + 144So QC2 = 400So QC = 20
So QC2 = QU2 + CU2 By Pythagoras Thm
Substitute
Square
Add
Simplify
ANSWER The radius of C = 20
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