11 3 arcs and chords lesson

Arcs and Chords Arcs and Chords

You will learn to identify and use the relationships amongarcs, chords, and diameters.

Nothing New!


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.


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


A

BC

The vertices of isosceles triangle ABC are located on R.

R

If BA AC, identify all congruent arcs.

BA AC


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.


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.


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


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

11 3 arcs and chords lesson

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