Transcript Geo Ch 10-3 – Arcs and Chords
Arcs and Chords
Chapter 10-3
• Recognize and use relationships between arcs and chords.
• Recognize and use relationships between chords and diameters.
• inscribed • circumscribed
Standard 7.0
involving Students prove and use theorems
the properties of parallel lines cut by a transversal, the properties of quadrilaterals, and t
he properties of circles.
(Key)
Standard 21.0
Students prove and solve problems regarding relationships among chords,
secants, tangents, inscribed angles, and inscribed and circumscribed polygons
of circles.
(Key)
Chord Theorems
• In the same circle or are circles, 2 minor arcs their corresponding chords are D B BC ED BC ED A E C
Prove Theorem 10.2
PROOF Write a two-column proof.
Given:
is a semicircle.
Prove:
Prove Theorem 10.2
Answer: Proof: Statements 1.
is a semicircle.
2. 3.
4. 5. Reasons 1.
Given
2.
Def. of semicircle
3.
In a circle, if 2 chords are , corr. minor arcs are .
4.
Def. of arcs
5.
Def. of arc measure
Answer: Statements 6.
7. 8.
Prove Theorem 10.2
9. 10. 11. Reasons 6.
7.
8.
Arc Addition Postulate Substitution Subtraction Property and simplify
9.
Division Property
10.
Def. of arc measure
11.
Substitution
PROOF Choose the best reason to complete the following proof. Given: Prove:
Proof: Statements 1.
2.
3.
4. Reasons 1.
Given
2.
In a circle, 2 minor arcs are , chords are .
3.
______
4.
In a circle, 2 chords are , minor arcs are .
A.
Segment Addition Postulate B.
Definition of
C.
Definition of Chord D.
Transitive Property A 0%
A.
A
B 0%
B.
0%
C.
D.
C
B C D
0% D
Inscribed Polygons
• If all the vertices of a polygon lie on the circle – The polygon is
inscribed
in the circle – The circle is
circumscribed
about the polygon
A regular hexagon is inscribed in a circle as part of a logo for an advertisement. If opposite vertices are connected by line segments, what is the measure of angle P in degrees?
Since connecting the opposite vertices of a regular hexagon divides the hexagon into six congruent triangles, each central angle will be congruent. The measure of each angle is 360 ÷ 6 or 60.
Answer:
60
ADVERTISING A logo for an advertising campaign is a pentagon that has five congruent central angles. Determine whether A.
yes B.
no C.
cannot be determined
1.
2.
3.
A B C
A 0% 0% B 0% C
Chord Theorems
• If the diameter of a circle is to a chord, the diameter bisects the chord and its arc AD DC A AB BC D B C
Radius Perpendicular to a Chord
Since radius is perpendicular to chord Arc addition Substitution Substitution Subtraction
Radius Perpendicular to a Chord
A radius perpendicular to a chord bisects it.
Def of seg bisector 8 10
Use the Pythagorean Theorem to find
WJ
.
Pythagorean Theorem
JK
= 8,
WK
= 10 Simplify.
Subtract 64 from each side.
Take the square root of each side.
6 10 Segment Addition Postulate
WJ
= 6,
WL
= 10 Subtract 6 from each side.
8
A.
35 B.
70 C.
105 D.
145
1.
2.
3.
4.
A B C D
A B C D
A.
15 B.
5 C.
10 D.
25
1.
2.
3.
4.
A
A B C
B
D
C D
Chord Theorems
• In the same circle or circles, 2 chords are they are equidistant from the center.
EF EG AB CD & AB CD A C B F E G D
Chords Equidistant from Center
24 Pythagorean Theorem 9 15 12 24
A.
B.
C.
D.
12 36 72 32 A 0%
A.
A
B 0%
B.
0%
C.
D.
C
B C D
0% D
A.
B.
C.
D.
12 36 72 32 A 0%
A.
A
B 0%
B.
0%
C.
D.
C
B C D
0% D
Chord Theorems Sample Problem • Solve for x + y AD = 3x + 7; DC = 5x +3 m AB = 4y + 8; m AEC = 96 C E D AD DC 3x + 7 = 5x + 3 4 = 2x A B AB BC 2=x AB ½ AC m AC = m AEC m AC = 96 4y + 8 = ½ (96) 4y + 8 = 48 4y = 40 y = 10