# Lesson 10.1 Parts of a Circle Today, we are going to…

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Lesson 10.1 Parts of a Circle Today, we are going to…
> identify segments and lines related to circles > use properties of tangents to a circle

Circle C C Diameter = _ radius

A chord is X Y N YX C A B AB BN

A secant is X Y C A B YX AB

A tangent is C Y X XY AB A B

Common Tangent Lines internal tangents

Common Tangent Lines external tangents

2 points of intersection
Two circles can intersect in 2, 1, or 0 points. Draw 2 circles that have 2 points of intersection

internally tangent circles
Draw two circles that have 1 point of intersection

externally tangent circles
Draw two circles that have 1 point of intersection

no point of intersection
concentric circles Draw two circles that have no point of intersection

9. What are the center and radius of circle A?

10. What are the center and radius of circle B?

11. Identify the intersection of the two circles.

12. Identify all common tangents of the
two circles.

m Ð ABC = A B C

Theorem 10.1 & 10.2 A line is tangent to a circle if and only if it is _____________ to the radius from the point of tangency. A B C

13. Find CA. C 7 D B 15 What is DA? A

14. Find x. C x 7 x x What is CA? B 8 6 16 15 A

15. Is AB a tangent? How do we test if 3 segments create a right triangle? C 10 7 26 B 6 24 15 A

16. Is AB a tangent? C 8 7 17 B 6 12 15 A

17. Find the slope of line t. A (3,0) and C (5, -1) t A Slope of AC? C

One endpoint is the point of tangency.
A tangent segment A B C One endpoint is the point of tangency.

Theorem 10.3 If 2 segments from the same point outside a circle are tangent to the circle, then they are congruent.

18. Find x. B 7x - 2 A C 3x + 8

19. Find x. B x2 + 25 A 50 C

Lesson 10.2 Arcs and Chords Today, we are going to…
> use properties of arcs and chords of circles

An angle whose vertex is the center of a circle is a
central angle. C A B

Minor Arc - Major Arc Major Arc ADB C D Minor Arc AB A B

Measures of Arcs C A B D 60˚ m AB =

Semicircle A B D E C m AED = m ABD = m AD

1. m BD 2. m DE 3. m FC 4. m BFD Find the measures of the arcs. D C
68˚ 52˚ ? B 100˚ E 53˚ F

AD and EB are diameters. 5. Find x, y, and z. E F D C A B x = x˚ 30˚

if and only if their chords are congruent.
Theorem 10.4 Two arcs are congruent if and only if their chords are congruent.

6. Find m AB B (3x + 11)° (2x + 48)° C D A

Theorem 10.5 & 10.6 A chord is a diameter if and only if it is a perpendicular bisector of a chord and bisects its arc.

7. Is AB a diameter? A B

8. Is AB a diameter? A B 8

9. Is AB a diameter? A B

Theorem 10.7 Two chords are congruent if and only if they are equidistant from the center.

10. Find CG. AB = 12 D G B A C F E DE = 12 x 7 6 ?

Lesson 10.3 Inscribed Angles Today, we are ALSO going to…
> use properties of inscribed angles to solve problems

An inscribed angle is an angle whose vertex is on the circle and whose sides contain chords of the circle.

Theorem 10.8 If an angle is inscribed, then its measure is half the measure of its intercepted arc. 2x x

1. Find x. x = 60° 120°

2. Find x. x = 140° 70°

Theorem 10.9 If 2 inscribed angles intercept the same arc, then the angles are congruent.

3. Find x and y. 45°

Inscribed Pentagon

4. DC is a diameter. Find x. C A D B

Theorem 10.10 If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle.

5. Find the values of x and y.
C A 42 D B

Theorem 10.11 If a quadrilateral is inscribed in a circle, then its
opposite angles are supplementary. 2 1 4 3 m 1 + m 3 = 180º m 2 + m 4 = 180º

6. Find the values of x and y.
80° 110°

7. Find the values of x and y.
100° 120°

Angle Relationships in Circles
Lesson 10.4 Angle Relationships in Circles Today, we are going to… > use angles formed by tangents and chords to solve problems > use angles formed by intersecting lines to solve problems

If a tangent and a chord intersect at a point on a circle, then...
Theorem 10.12 If a tangent and a chord intersect at a point on a circle, then... GSP

… the measure of each angle formed is half
Theorem 10.12 … the measure of each angle formed is half the measure of its intercepted arc.

B C 2 1 A

1. Find m 1 and m 2. B C 100° 2 1 A

2. Find and mACB and mAB 95° A B C

5x° A B C (9x + 20)˚ 3. Find x

If 2 chords intersect inside a circle, then…
Theorem 10.13 If 2 chords intersect inside a circle, then… A C 1 B D

…the measure of the angle is half the sum of the intercepted arcs.
1 B D …the measure of the angle is half the sum of the intercepted arcs.

4. Find x. 100° A B C D 120°

5. Find x. 130° A B C D 160°

6. Find x. A C 80° 90° B D

7. Find x. A B C D 100° 120°

8. Find x. A B C D 74° 52° Do you notice a pattern?

If a tangent and a secant,
Theorem 10.14 If a tangent and a secant, two tangents, or two secants intersect outside a circle, then… A C D 1

If a tangent and a secant,
Theorem 10.14 If a tangent and a secant, two tangents, or two secants intersect outside a circle, then… A B C 1

If a tangent and a secant,
Theorem 10.14 If a tangent and a secant, two tangents, or two secants intersect outside a circle, then… A B C D 1

A B C D 1 …the measure of the angle is half the difference of the intercepted arcs.

9. Find x. A B C D 20° 80°

10. Find x. 24° 90° A B C D

11. Find x. 200°

12. Find x. A C D 135°

13. Find x. 100° 2 3 60° 100° 1 100°

Lesson 10.5 Segment Lengths in Circles
Today, we are going to… > find the lengths of segments of chords, tangents, and secants

Theorem 10.15 If 2 chords intersect inside a circle, then the product of their “segments” are equal.

a c d b a · b = c · d

1. Find x. 6 x 8 4

2. Find x. 3x 3 18 2x

3. Find x. 2x 18 x 4

Theorem 10.16 If 2 secant segments share the same endpoint outside a circle, then… GSP

…one secant segment times its external part equals the other secant segment times its external part.

c a b d a · c = b · d

5 x 4 6 3. Find x.

9 10 x 20 4. Find x.

Theorem 10.17 If a secant segment and a tangent segment share an endpoint outside a circle, then…

…the length of the tangent segment squared equals the length of the secant segment times its external part.

a b d a · a = b · d a2 = b · d

5 4 x 5. Find x.

15 x 10 6. Find x.

15 x 10 6. Find x.

x 20 31 7. Find x.

3 4 8 x 8. Find x.

10 x 8 9. Find x.

Lesson 10.6 Equations of Circles Today, we are going to…
> write the equation of a circle

Standard Equation for a Circle with Center: (0,0) Radius = r

1. Write an equation of the circle.

2. Write an equation of the circle.

Standard Equation for a Circle with Center: (h,k) Radius = r

3.Write an equation of the circle.

4.Write an equation of the circle.

Graph (x – 3)2 + (y + 2)2 = 9 Center? Radius =

Identify the center and radius of
the circle with the given equation. 5. (x – 1)2 + (y + 3)2 = 100 Center: (1, -3) radius = 10 6. x2 + (y - 7)2 = 8 Center: (0, 7) radius ≈ 2.83 7. (x + 1)2 + y2 = ¼ radius = ½ Center: (-1, 0)

Write the standard equation of the circle with a center of (5, -1) if a point on the circle is (1,2).

8. Write the standard equation of. the circle with a center of (-3, 4)
8. Write the standard equation of the circle with a center of (-3, 4) if a point on the circle is (2,-5).

Is (-2,-10) on the circle (x + 5)2 + (y + 6)2 = 25?

9. Is (0, - 6) on the circle (x + 5)2 + (y – 5)2 = 169?

10. Is (2, 5) on the circle (x – 7)2 + (y + 5)2 = 121?

< > =

Would the point be inside the circle, outside the circle, or on the circle?
(x – 13)2 + (y - 4)2 = 100 11. (11, 13) 12. (6, -5) 13. (19, - 4)

Circumference and Area of Circles
Lessons 11.4 & 11.5 Circumference and Area of Circles Today, we are going to… > find the length around part of a circle and find the area of part of a circle

Circumference

Arc Length = A B

1. Find the length of AB A B 50° 7 cm

2. Find the radius A 10 cm 85° B

3. Find the circumference.

Area

A region bound by two radii & their intercepted arc.
Sector of a circle A region bound by two radii & their intercepted arc. A slice of pizza!

Area of a Sector =

3. Find the area of the sector.
50° B 7 cm

A B 100° 4. Find the radius.

3. Find the area.

Workbook P. 211 (1 – 10) P. 215 (1 – 6)

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