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10.1 Tangents to Circles (595) Notes #4-8 Date: ______

circle: locus of points equidistant from a point (center) in a plane.

radius: distance from the center to a point on the circle, the segments whose endpoints are the center and a point on the circle.

Congruent circles: circles with the same radius length.

Chord: segment whose endpoints are points on the circle.

Diameter: distance across a circle through the center (chord that passes through the center of the circle).

Secant: a line in the plane that intersects a circle in two points.

Tangent: a line/line segment in the plane that intersects the circle in exactly 1 point.

Ex.1 The segment named is best described as a chord, a secant, a tangent, a diameter, or a radius. (* Alphabetical)

a. AH b. EI c. DF d. CE e. Name a secant:

Tangent circles: coplanar circles that intersect in one point. Concentric circles: Coplanar circles that have a common center. Ex.2 Tell whether the common Ex.3 Give the center & radius of tangent is internal or external. each circle. Describe the intersection of the 2 circles & describe all common tangents.

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Ex. 4 Is CE tangent to circle D? Explain. Ex. 5 You are standing 14 feet from a water tower. The distance from you

to a point of tangency on the tower is 28 feet. What is the radius of the water tower?

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Ex. 6 Given: CB is tangent to circle A at B. CD is tangent to circle A at D. Prove: ΔABC ≅ ΔADC. Statements Reasons 1. CB is tangent to A at B 1. Given CD is tangent to A at D 2. BC ≅ DC 2. tangents ≅ 3. AB ≅ AD 3. radii of circle ≅ 4. AC ≅ AC 4. Reflexive 5. ΔABC ≅ ΔADC 5. SSS ≅ Ex. 7 AB is tangent to circle C at B. AD is tangent to circle C at D. Find the

value of x. Ex.8 Find the polygon’s perimeter:

14 ft

4 ft

6 ft

7 ft

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10.2 Arcs and Chords (603) Notes #4-9 Date: ______ Major arcs & semicircles must be named by their endpoints & a point on the arc. The measure of a minor arc is equal to the measure of its central angle. The measure of a major arc is defined as 360° minus the measure of its associated minor arc. Ex.1 Find the measure of each arc. Ex.2 Find the measure of each arc. a. CD a. BD b. CBD b. BED c. BCD c. BE Congruent arcs: two arcs of the same circle or congruent circles that have the same measure Ex.3 Find the measures of arc AB & arc CD. Are the arcs congruent?

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Ex. 4 Use Theorem 10.4 to find the measure of arc BC. Ex. 5 Locate the center of the following circle using the chords shown.

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Ex. 6 Find the measure of arc MN. a) b) Ex. 7 AB=12, DE=12, and CE=7. Find CG.

R.1 Graph: y – 3 = − 5

12(x + 6), y = −3

2x – 6 & 6x – 9y = 54.

Perimeter: Area: Classify the enclosed triangle: ____________ & ____________

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10.3 Inscribed Angles (613) Notes #4-10 Date: ______ Ex.1 Find the measure of the arc or angle. a) arc ADC b) arc AC c) m∠ABC Ex.2 Find the measures of the angles: m∠ABE = m∠ACE = m∠ADE =

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Ex.3 It is given that m∠B= 44°. What is m∠C? Ex. 4 Find the value of each variable. c.

(6x2 + 98)°

(39x + 61)°

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Ex. 5 In the diagram, ABCD is inscribed in P. Find the measure of each angle. AMC 10 – 2008 #16

Ex.6 B Ex.7

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10.4 Other Angle Relationships in Circles (621) Notes #4-11 Date: ______ Ex.1 Line m is tangent to the circle. Find m arc RST. Ex.2 BC is tangent to the circle. Find m∠CBD.

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Ex.3 Find the value of x. a) b) Ex. 4 Find the value of x. a) b) c) Ex. 5 Solve for the obtuse angle. Ex.6 Solve for x.

143°

x

70°

x°

(3x + 9)°

(5x – 1)°

12x°

(190 – 13x)°

(4x2)°

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10.5 Segment Lengths in Circles (629) Notes #4-12 Date: ______ Similar triangles! Ex.1 Find the value of x. a) b) Ex.2 Find the value of x. a) b)

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Ex. 3 Find the value of x. a) b) Ex.4 You are standing 18 feet from a circular display case. The distance

from you to a point of tangency on the case is about 32 feet. Estimate the radius of the display case.

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10.6 Equations of Circles (636) Notes #4-13 Date: ______ Write an equation for the distance r: Solve for r2: What if (h, k) is the origin? Write the standard equation of a circle with the given center & …

Ex.1 (-5,0) & radius 2.5 Ex.2 (4,-3) & point (2,1) on the circle Ex.3 a) Graph (x – 3)2 + (y + 1)2 = 4. b) Write the equation of the circle. Ex. 4 The equation of a circle is (x – 13)2 + (y – 6)2 = 9. Three points are

located as follows: A(5,6), B(14,8), and C(16, 6). Where are the three points located relative to the circle?

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Challenge (completing the square):

Multiply: (x + 11)2 =

Factor: x2 + 6x + 9 x2 – 10x + 25

(x + )2 (x – )2

What constant would result in a perfect square trinomial?

x2 + 14x + x2 – 2x +

Ex.5 Convert the general form to the standard equation of the circle: a) x2 + y2 + 4x – 6y – 23 = 0 b) x2 + y2 – 2y – 15 = 0 NPE 2009 #8

Ex.6 A rectangle with sides parallel to the coordinate axes is inscribed in x2 + y2 = 169. The area of the rectangle is 240 and (a, b) is the point in the first quadrant where the rectangle intersects the circle. What is a + b?

Derive the Quadratic Formula from 0 = ax2 + bx + c.

half squared

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10.7 Locus (642) Notes #4-14 Date: ______

Locus: set of all points that satisfy a set of given conditions. Ex.1 Draw a line k. Draw and describe the locus of points, in the plane,

that are 1 inch from the line. Ex.2 Given ΔABC, what is the locus of points in the plane equidistant from

AB and AC and also equidistant from A and B? Ex.3 What are the possible loci of points 2 cm from point X and 3 cm from

point Y?

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Ex. 4 You are given readings from 3 seismographs. At A(2,4), the epicenter is 2 miles away. At B(2, -0.5), the epicenter is 2.5 miles away. At C(-1,1) the epicenter is 3 miles away. Where is the epicenter? Ex.5 What is the locus of points Ex.6 What is the locus of all points equidistant from the lines: in the plane equidistant from y = 3

5x + 2 and 3x – 5y = -50? A(1, -2) & B(3, -2) & 5 units

from B? Ex.7 What is the locus of points equidistant from x = -3 and y = 2?