Post on 12-Jan-2015
description
Arcs and Chords Arcs and Chords
You will learn to identify and use the relationships amongarcs, chords, and diameters.
Nothing New!
Arcs and Chords Arcs and Chords
BD
S
C
P
A
In circle P, each chord joins two points on a circle.
Between the two points, an arc forms along the circle.
vertical angles
By Theorem 11-3, AD and BC are congruentbecause their corresponding central angles are_____________, and therefore congruent.
By the SAS Theorem, it could be shown thatΔAPD ΔCPB.
Therefore, AD and BC are _________.congruent
The following theorem describes the relationship between two congruentminor arcs and their corresponding chords.
Arcs and Chords Arcs and Chords
Theorem11-4
In a circle or in congruent circles, two minor arcs are congruentif and only if (iff) their corresponding ______ are congruent.
B
D
C
A
chords
AD BCiff
AD BC
Arcs and Chords Arcs and Chords
A
BC
The vertices of isosceles triangle ABC are located on R.
R
If BA AC, identify all congruent arcs.
BA AC
Arcs and Chords Arcs and Chords
Step 1) Use a compass to draw circle on a piece of patty paper. Label the center P. Draw a chord that is not a diameter. Label it EF.
Step 2) Fold the paper through P so that E and F coincide. Label this fold as diameter GH.
E
F
P
G
H
Q1: When the paper is folded, how do the lengths of EG and FG compare?
Q2: When the paper is folded, how do the lengths of EH and FH compare?
Q3: What is the relationship between diameter GH and chord EF?
EG FG
EG FG
They appear to be perpendicular.
Arcs and Chords Arcs and Chords
Theorem11-5
In a circle, a diameter bisects a chord and its arc if and only if(iff) it is perpendicular to the chord.
PR
D
C
B
AAR BR and AD BD
iff
CD AB
Like an angle, an arc can be bisected.
Arcs and Chords Arcs and Chords
B
C
A
K
D
7
Find the measure of AB in K.
AB=2 ( DB ) Theorem 11-5
AB=2 (7 )AB= 14
Substitution
Arcs and Chords Arcs and Chords
M
K
L
K
N6
Find the measure of KM in K if ML = 16.
( K M )2= ( KN )2+ ( MN ) 2Pythagorean Theorem
( K M )2= (6 )2+ (8 )2 Given; Theorem 11-5
( KM )2=36+ 64
( K M )2= 100
√ ( KM )2=√100K M = 10