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Arcs Tangents Angles Sphere Circumference What is Unit 3 about? We will learn to use arcs, angles, and segments in circles to solve real life problems. We will learn how to find the measure of angles related to a circle. We will find circumference and area of figures with circles. We will also find the surface and volume of spheres.

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Crop Circles Whether you think crop circles are made by little green men from space or by sneaky earthling geeks, you've got to admit that they are pretty dang cool... And whoever is making them knows a ton of geometry!

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**Properties of Tangents**

Tuesday, April 11, 2017 Essential Question: How do we identify segments and lines related to circles and how do we use properties of a tangent to a circle? Lesson 6.1 M2 Unit 3: Day 1

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Warm Ups 1. What measure is needed to find the circumference or area of a circle? ANSWER radius or diameter 2. Find the radius of a circle with diameter 8 centimeters. ANSWER 4 cm 3. A right triangle has legs with lengths 5 inches and 12 inches. Find the length of the hypotenuse. ANSWER 13 in. 4. Solve 6x + 15 = 33. 5. Solve (x + 18)2 = x ANSWER 7 ANSWER 3

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Circle The set of all points in a plane that are equidistant from a given point, called the center. E F Circle Tangent Chord P Diameter Radius D A Secant C B Name of the circle: ʘ P

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Definition Radius – a segment from the center of the circle to any point on the circle. Diameter – a chord that contains the center of the circle. C A B

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Definition Chord – a segment whose endpoints are points on the circle.

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Definition Secant – a line that intersects a circle in two points.

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Definition Tangent – a line in the plane of a circle that intersects the circle in exactly one point. P Point of tangency O

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**A little extra information**

The word tangent comes from the Latin word meaning to touch The word secant comes from the Latin word meaning to cut.

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EXAMPLE 1 1. Tell whether the line or segment is best described as a chord, a secant, a tangent, a diameter, or a radius. tangent diameter chord radius

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**Identify special segments and lines**

EXAMPLE 2 Identify special segments and lines 2. Tell whether the line, ray, or segment is best described as a radius, chord, diameter, secant, or tangent of ʘ C. AC a. b. AB DE c. AE d. SOLUTION is a radius because C is the center and A is a point on the circle. AC a. b. AB is a diameter because it is a chord that contains the center C. c. DE is a tangent ray because it is contained in a line that intersects the circle at only one point. d. AE is a secant because it is a line that intersects the circle in two points.

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EXAMPLE 3 Find lengths in circles in a coordinate plane 3. Use the diagram to find the given lengths. a. Radius of ʘ A b. Diameter of ʘ A Radius of ʘ B c. Diameter of ʘ B d. SOLUTION a. The radius of ʘ A is 3 units. b. The diameter of ʘ A is 6 units. c. The radius of ʘ B is 2 units. d. The diameter of ʘ B is 4 units.

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GUIDED PRACTICE EXAMPLE 4 4. Find the radius and diameter of ʘ C and ʘ D. SOLUTION a. The radius of ʘ C is 3 units. b. The diameter of ʘ C is 6 units. c. The radius of ʘ D is 2 units. d. The diameter of ʘ D is 4 units.

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Definitions Common tangent – a line or segment that is tangent to two coplanar circles Common internal tangent – intersects the segment that joins the centers of the two circles Common external tangent – does not intersect the segment that joins the centers of the two circles

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**5. Tell whether the common tangents are internal or external.**

EXAMPLE 5 5. Tell whether the common tangents are internal or external. a. b. common internal tangents common external tangents

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Draw common tangents EXAMPLE 6 6. Tell how many common tangents the circles have and draw them. a. b. c. SOLUTION a. 4 common tangents 3 common tangents b. c. 2 common tangents

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**Perpendicular Tangent Theorem 6.1**

In a plane, if a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.

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Tangent Theorems Create right triangles for problem solving.

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**Write the binomial twice. Multiply.**

EXAMPLE 8 Find the radius of a circle 7. In the diagram, B is a point of tangency. Find the radius r of ʘC. SOLUTION You know that AB BC , so △ ABC is a right triangle. You can use the Pythagorean Theorem. AC2 = BC2 + AB2 Pythagorean Theorem (r + 50)2 = r Substitute. (r + 50)(r + 50) = r Write the binomial twice. Multiply. r2 + 50r +50r = r r r = r Combine Like Terms. 100r = 3900 Subtract from each side. r = 39 ft . Divide each side by 100.

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**Subtract from each side.**

EXAMPLE 8 8. ST is tangent to ʘ Q. Find the value of r. SOLUTION You know from Theorem 10.1 that ST QS , so △ QST is a right triangle. You can use the Pythagorean Theorem. QT2 = QS2 + ST2 Pythagorean Theorem (r + 18)2 = r Substitute. r2 + 36r = r Multiply. 36r = 252 Subtract from each side. r = 7 Divide each side by 36.

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**Perpendicular Tangent Converse**

In a plane, if a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is tangent to the circle.

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GUIDED PRACTICE EXAMPLE 10 9. Is DE tangent to ʘ C? ANSWER Yes – The length of CE is 5 because the radius is 3 and the outside portion is 2. That makes ∆CDE a Right Triangle. So DE and CD are

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**Verify a tangent to a circle**

EXAMPLE 7 10. In the diagram, PT is a radius of ʘ P. Is ST tangent to ʘ P ? SOLUTION Use the Converse of the Pythagorean Theorem. Because = 372, △ PST is a right triangle and ST PT . So, ST is perpendicular to a radius of ʘ P at its endpoint on ʘ P. ST is tangent to ʘ P.

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**Congruent Tangent Segments Theorem 6.2**

If two segments from the same exterior point are tangent to a circle, then they are congruent.

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**Tangent segments from the same point are**

11. RS is tangent to ʘ C at S and RT is tangent to ʘC at T. Find the value of x. SOLUTION RS = RT Tangent segments from the same point are 28 = 3x + 4 Substitute. 8 = x Solve for x.

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12.

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Homework Page 187 # 18 – 24 all Page 188 # 1 – 10.

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10.1 Tangents to Circles.

10.1 Tangents to Circles.

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