Download 11-2 Chords and Arcs

11-2 CHORDS AND ARCS (p. 590-596)
A chord is a segment within a circle whose endpoints are on the circle.
C
B
CB is a chord in Circle A
A
An arc that is related to a chord has the same endpoints as the chord. Name the related
arc to chord CB in the above figure.
A central angle is an angle whose vertex is at the center of the circle and whose two sides
are radii. A central angle can intercept both a chord and an arc. When a central angle
intercepts a chord or arc, the sides of the central angle intersect the endpoints of the chord
or arc.
Example: Name the central angle in the following diagram. Name the chord that the
central angle intercepts. Name the arc that the central angle intercepts.
C
T
A
In a circle or in circles that are congruent to each other, there is a relationship among
central angles, chords, and arcs. This relationship is expressed in the following super
theorem.
Theorem 11-4
Within a circle or in congruent circles:
1. Congruent central angles have (or intercept) congruent chords.
2. Congruent chords have congruent arcs.
3. Congruent arcs have congruent central angles.
These three relationships can be summarized in a triangular diagram. In summary, if one
pair of items are congruent, then two other pairs of items are congruent.
Congrue nt
Central
Angles
Congrue nt
Arcs
Congrue nt
Chords
In reality, the rays are actually lines (or double-headed arrows). You can really go
clockwise (like above) or counterclockwise.
Example: Prove part 1 of Theorem 11-4 by sketching congruent central angles in a
circle. Show that the intercepted chords are congruent by using congruent triangles and
CPCTC.
Example: Have students answer #24 on p. 594. This is a flow proof of part 2 of
Theorem 11-4. Recall that in a circle, congruent central angles intercept congruent arcs.
Example: In the following diagram of circle O, OX bisects AOB. Give three different
conclusions that you can reach and a reason for each conclusion.
A
X
O
B
Conclusion
Reason
1.
2.
3.
Another theorem describes the relationship between two chords and their distances from
the center of a circle.
Theorem 11-5
Within a circle or in congruent circles
1. Chords equidistant from the center are congruent.
2. Congruent chords are equidistant from the center.
Note that the two parts of Theorem 11-5 are converses of each other.
On p. 591, examine the proof of part 1 of Theorem 11-5 and note these special points
from the following statement numbers.
4. Two triangles are proved congruent by HL.
5. What is another way to remember the isosceles triangle theorem?
9. Our first theorem of this section is used in the proof of this theorem.
We will not be concerned with the proof of part 2 of Theorem 11-5 since it is very similar
to that of part 1.
Example:
In the following diagram of Circle O, OX = 5, OY = 5,
OX  AB, and OY  CD. Also, AX  BX , AX  2x  3, and CD  7x - 20. Set up and
solve an equation to find x.
A
X
B
O
D
C
Y
Do 2 on p. 591.
Let us recall a theorem from Chapter 5, Section 2.
X
A
B
If XA = XB (or XA  XB) , then X lies on a special line with respect to AB. What
special line is this?
We learned this as Theorem 5-3.
If a point is equidistant from the endpoints of a segment, then it is on the
perpendicular bisector of the segment.
Theorem 11-6
In a circle, a “diameter” that is perpendicular to a chord bisects the chord and the
arc that has the same endpoints as this chord.
In the following diagram of circle O, MP  JK. Answer the following questions.
M
J
V
K
O
P
1. Why is OJ  OK ? So, O is equidistant from J and K.
2. What special “line” is MP with respect to JK ? Thus, why is JV  KV ?
3. By what method is JOV  KOV? Thus, why is JOM  KOM?
4. If JOM  KOM, why is arc JM congruent to arc MK?
Note: You will more formally prove this theorem in homework problem #25.
The next theorem is the converse of Theorem 11-6. Since its proof is similar to that for
Theorem 11-6, we will not prove it.
Theorem 11-7
In a circle, a “diameter” that bisects a chord (that is not a diameter) is
perpendicular to the chord.
The final theorem in this trio of theorems that deal with “lines” going through the center
of a circle is somewhat of a combination of the first two theorems (11-6 and 11-7).
Theorem 11-8
In a circle, the perpendicular bisector of a chord contains the center of the circle.
Here is a quick way to summarize Theorems 11-6 through 11-8. Basically, if you have a
“line” satisfying two of the following three conditions, then the third condition
automatically follows.
Pe rpe ndicular to chord
Goes through center
Bisects chord
Example: In the following diagram of circle P, find x. Why is PK  AB ? Leave your
answer in simplest radical form.
A
14
P
10
K
x 10
B
Example: P and Q are points on circle O. The distance from O to PQ is 15 cm and
PQ = 16 cm. Find the radius of circle O. Make a helpful sketch.
Do 3 on p. 593. What simple operation can you sue to answer 3b?
Homework p. 593-596: 2,6,7,9,10,12,16-18,21,25-27,31,41,45,48
21. 5-12-13
12 cm
26.
x
4
Solve x 2  16  64
4
x  13.9 cm
31. AB and CD are perpendicular bisectors of each other. Use 5-12-13. 2(5) = 10 cm