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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
C Circle C Diameter = _ radius
C A chord is YX AB A B X Y N BN
C A secant is A B X Y YX AB
C A tangent is AB A B Y X XY
internal tangents Common Tangent Lines
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
concentric circles Draw two circles that have no point of intersection
9. What are the center and radius of circle A? Center: Radius =
10. What are the center and radius of circle B? Center: Radius =
11. Identify the intersection of the two circles.
12. Identify all common tangents of the two circles.
m 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
7 13. Find CA. 15 D C B A What is DA?
7 14. Find x. 15 x 6 C B A xx 16 8 What is CA?
7 15 6 C B A 26 10 24 How do we test if 3 segments create a right triangle? 15. Is AB a tangent?
7 15 6 C B A 17 8 12 16. Is AB a tangent?
17. Find the slope of line t. A C A (3,0) and C (5, -1) Slope of AC? Slope of line t? t
C A tangent segment A B 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.
7x - 2 3x + 8 18. Find x. A C B
x 2 + 25 50 19. Find x. A C B
Lesson 10.2 Arcs and Chords Today, we are going to… > use properties of arcs and chords of circles
C An angle whose vertex is the center of a circle is a central angle. A B
C Minor Arc - Major Arc A B D Minor Arc AB Major Arc ADB
C A B D 60˚ m AB = Measures of Arcs
C Semicircle m AED = m ABD = m AD A B D E
Find the measures of the arcs. 1. m BD 2. m DE 3. m FC 4. m BFD D E F B C 100˚ 52˚ 68˚ 53˚ ?
AD and EB are diameters. F A B D E C 5. Find x, y, and z. 30˚ z˚z˚ x˚x˚ y˚y˚ x = y = z =
Theorem 10.4 Two arcs are congruent if and only if their chords are congruent.
(2x + 48)° (3x + 11)° B A D C 6. Find m AB
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 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.
AB = 12 10. Find CG. DE = 12 7 D G B A C F E 6 x ?
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. xx 2x
1. Find x. x°x° 120° x = 60°
2. Find x. x°x° 70° x = 140°
Theorem 10.9 If 2 inscribed angles intercept the same arc, then the angles are congruent.
3. Find x and y. y°y° 45° x°x°
x°x° A D C B 4. DC is a diameter. Find x.
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. x°x° y°y° A 42 D C 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. x°x° 110° 80° y°y°
7. Find the values of x and y. x°x° 120° 100° y°y°
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
Theorem 10.12 If a tangent and a chord intersect at a point on a circle, then... GSP
Theorem 10.12 … the measure of each angle formed is half the measure of its intercepted arc.
1 A B C 2
1 A B C 2 1. Find m 1 and m 2. 100°
2. Find and mACB and mAB 95° A B C
3. Find x 5x° A B C (9x + 20)˚
Theorem 10.13 If 2 chords intersect inside a circle, then… A B C D 1
B C A D 1 …the measure of the angle is half the sum of the intercepted arcs.
A B C D x°x° 4. Find x. 100° 120°
A B C D x°x° 5. Find x. 130° 160°
A B C D x°x° 6. Find x. 80° 90° y°y°
A B C D x°x° 7. Find x. 100° 120°
A B C D x°x° 8. Find x. 52° 74° Do you notice a pattern?
Theorem 10.14 If a tangent and a secant, two tangents, or two secants intersect outside a circle, then… A C D 1
Theorem 10.14 If a tangent and a secant, two tangents, or two secants intersect outside a circle, then… A B C 1
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. 20° 80° A B C D x°x°
10. Find x. 24° 90° A B C D x°x°
11. Find x. 200° x°x°
A C D 12. Find x. 135° x°x°
13. Find x. 100° 3 2 1 60°
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 · b = c · da · b = c · d a b c d
1. Find x. 6 8 4 x
2. Find x. 3x 18 2x 3
3. Find x. 2x 18 x 4
Theorem 10.16 If 2 secant segments share the same endpoint outside a circle, then… GSP GSP
…one secant segment times its external part equals the other secant segment times its external part.
a · c = b · d b a c d
3. Find x. 5 x 4 6
4. Find x. 9 10 x 20
Theorem 10.17 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 · a = b · d d b a a2 = b · da2 = b · d
5 4 x 5. Find x.
15 x 10 6. Find x.
Quadratic Formula? ♫♪♫♪♫♪♫♪♫♪♫♪♫♪♫♪♫♪♫♪♫♪♫♪
15 x 10 6. Find x.
x 20 31 7. Find x.
8. Find x. 3 4 8 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. C = r =
4.Write an equation of the circle. C = r =
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 6. x 2 + (y - 7) 2 = 8 7. (x + 1) 2 + y 2 = ¼ Center: (1, -3)radius = 10 Center: (0, 7)radius ≈ 2.83 Center: (-1, 0) radius = ½
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) 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)
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
Arc Length = A B
A B 50° 7 cm 1. Find the length of AB
A B 85° 10 cm 2. Find the radius
3. Find the circumference.
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. A B 50° 7 cm
4. Find the radius. A B 100°
3. Find the area.
Workbook P. 211 (1 – 10) P. 215 (1 – 6)
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