1. Lesson 8-4: Arcs and Chords
Warm-Up #34 Tuesday, 5/17
Lesson 8-4: Arcs and Chords

2. Lesson 8-4: Arcs and Chords
Warm-Up #35 Wednesday, 5/18
1. Find the perimeter of the polygon
2. Find the distance between the centers of the pulleys.
3. The radius of Earth is about 6400 km. Find the distance d, given h= 1km
Lesson 8-4: Arcs and Chords

3. Lesson 8-4: Arcs and Chords
Arcs and Chords page 1 and 2
Lesson 8-4: Arcs and Chords

4. Arcs
and Chords

5. Definition
Central angle – an angle whose vertex is the center of a circle.

6. Definitions Minor arc – Part of a circle that measures less than 180°
Major arc – Part of a circle that measures between 180° and 360°.
Semicircle – An arc whose endpoints are the endpoints of a diameter of the circle.
Note : major arcs and semicircles are named with three points and minor arcs are named with two points

7. Definitions Measure of a minor arc – the measure of its central angle
Measure of a major arc – the difference between 360° and the measure of its associated minor arc.

9. Arc Addition Postulate
The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.

10. Example 1
Find the measure of each arc.
70°
360° - 70° = 290°
180°

11. Example 2
Find the measures of the red arcs. Are the arcs congruent?

12. Example 3
Find the measures of the red arcs. Are the arcs congruent?

13. Arcs and Chords Theorem
In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.

14. Perpendicular Diameter Theorem
If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.

15. Perpendicular Diameter Converse
If one chord is a perpendicular bisector of another chord which must pass through the center of the circle, then the first chord is a diameter.

16. Congruent Chords Theorem
In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center.

17. Example 4

18. Example 3A: Applying Congruent Angles, Arcs, and Chords
TV  WS. Find mWS.
TV  WS
 chords have  arcs.
mTV = mWS
Def. of  arcs
9n – 11 = 7n + 11
Substitute the given measures.
2n = 22
Subtract 7n and add 11 to both sides.
n = 11
Divide both sides by 2.
mWS = 7(11) + 11
Substitute 11 for n.
= 88°
Simplify.

19. Example 3B: Applying Congruent Angles, Arcs, and Chords
C  J, and mGCD  mNJM. Find NM.
GCD  NJM
GD  NM
 arcs have  chords.
GD  NM
GD = NM
Def. of  chords

20. Example 3B Continued
C  J, and mGCD  mNJM. Find NM.
14t – 26 = 5t + 1
Substitute the given measures.
9t = 27
Subtract 5t and add 26 to both sides.
t = 3
Divide both sides by 9.
NM = 5(3) + 1
Substitute 3 for t.
= 16
Simplify.

21. Check It Out! Example 3a
PT bisects RPS. Find RT.
RPT  SPT
mRT  mTS
RT = TS
6x = 20 – 4x
10x = 20
Add 4x to both sides.
x = 2
Divide both sides by 10.
RT = 6(2)
Substitute 2 for x.
RT = 12
Simplify.

22. Check It Out! Example 3b
Find each measure.
A  B, and CD  EF. Find mCD.
mCD = mEF
 chords have  arcs.
Substitute.
25y = (30y – 20)
Subtract 25y from both sides. Add 20 to both sides.
20 = 5y
4 = y
Divide both sides by 5.
CD = 25(4)
Substitute 4 for y.
mCD = 100
Simplify.

23. Example 4: Using Radii and Chords
Find NP.
Step 1 Draw radius RN.
RN = 17
Radii of a  are .
Step 2 Use the Pythagorean Theorem.
SN2 + RS2 = RN2
SN = 172
Substitute 8 for RS and 17 for RN.
SN2 = 225
Subtract 82 from both sides.
SN = 15
Take the square root of both sides.
Step 3 Find NP.
NP = 2(15) = 30
RM  NP , so RM bisects NP.

24. Check It Out! Example 4
Find QR to the nearest tenth.
Step 1 Draw radius PQ.
PQ = 20
Radii of a  are .
Step 2 Use the Pythagorean Theorem.
TQ2 + PT2 = PQ2
TQ = 202
Substitute 10 for PT and 20 for PQ.
TQ2 = 300
Subtract 102 from both sides.
TQ  17.3
Take the square root of both sides.
Step 3 Find QR.
QR = 2(17.3) = 34.6
PS  QR , so PS bisects QR.

25. Lesson 8-4: Arcs and Chords
Try Some Sketches:
Draw a circle with a chord that is 15 inches long and 8 inches from the center of the circle.
Draw a radius so that it forms a right triangle.
How could you find the length of the radius?
Solution:
∆ODB is a right triangle and
8cm
15cm O A B D x
Lesson 8-4: Arcs and Chords

26. Lesson 8-4: Arcs and Chords
Try Some Sketches:
Draw a circle with a diameter that is 20 cm long.
Draw another chord (parallel to the diameter) that is 14cm long.
Find the distance from the smaller chord to the center of the circle.
Solution:
10 cm
20cm O A B D C
∆EOB is a right triangle.
OB (radius) = 10 cm
14 cm E x
7.1 cm
Lesson 8-4: Arcs and Chords

27. Lesson Quiz: Part I
1. The circle graph shows the types of cuisine available in a city. Find mTRQ.
158.4

28. Lesson Quiz: Part II
Find each measure.
2. NGH
139
3. HL 21

29. Lesson Quiz: Part III
4. T  U, and AC = Find PL to the nearest tenth.
 12.9