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MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. 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. ArcsTangentsAnglesSphereCircumference

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MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. 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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MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. Properties of Tangents Essential Question: How do we identify segments and lines related to circles and how do we use properties of a tangent to a circle? M2 Unit 3: Day 1 Saturday, April 18, 2015 Lesson 6.1

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MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. 1.What measure is needed to find the circumference or area of a circle? 2.Find the radius of a circle with diameter 8 centimeters. ANSWER radius or diameter 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. ANSWER 3 5.Solve (x + 18) 2 = x ANSWER 7 Warm Ups

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MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.

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MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. Circle A C Secant D E Chord F Tangent P Name of the circle: ʘ P DiameterRadius B Circle The set of all points in a plane that are equidistant from a given point, called the center.

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MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. Definition Diameter – a chord that contains the center of the circle. Radius – a segment from the center of the circle to any point on the circle. A B C

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MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. Definition Chord – a segment whose endpoints are points on the circle.

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MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. Definition Secant – a line that intersects a circle in two points.

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MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. Definition Tangent – a line in the plane of a circle that intersects the circle in exactly one point. O P Point of tangency

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MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. 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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MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. 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 EXAMPLE 1

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MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. EXAMPLE 2Identify 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. SOLUTION is a radius because C is the center and A is a point on the circle. AC a. b. ABis a diameter because it is a chord that contains the center C. b. AB c. DE is a tangent ray because it is contained in a line that intersects the circle at only one point. DE c. d. AE is a secant because it is a line that intersects the circle in two points. AE d.

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MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. EXAMPLE 3Find lengths in circles in a coordinate plane b. Diameter of ʘ ARadius of ʘ B c. Diameter of ʘ B d. 3. Use the diagram to find the given lengths. a. Radius of ʘ A 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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MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. SOLUTION GUIDED PRACTICE 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. 4. Find the radius and diameter of ʘ C and ʘ D. EXAMPLE 4

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MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. 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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MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. 5. Tell whether the common tangents are internal or external. a. b. common internal tangentscommon external tangents EXAMPLE 5

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MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. EXAMPLE 6 Draw common tangents 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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MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. 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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MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. Tangent Theorems Create right triangles for problem solving.

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MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. EXAMPLE 8 Find the radius of a circle 7. In the diagram, B is a point of tangency. Find the radius r of ʘ C. SOLUTION You know that AB BC, so △ ABC is a right triangle. You can use the Pythagorean Theorem. AC 2 = BC 2 + AB 2 (r + 50) 2 = r r r = r r = 3900 r = 39 ft. Pythagorean Theorem Substitute. Write the binomial twice. Subtract from each side. Divide each side by 100. (r + 50)(r + 50) = r r r +50r = r Combine Like Terms. Multiply.

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MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. 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. r r = r r = 252 r = 7 Multiply. Subtract from each side. Divide each side by 36. QT 2 = QS 2 + ST 2 (r + 18) 2 = r Pythagorean Theorem Substitute. EXAMPLE 8

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MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. 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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MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. GUIDED PRACTICE 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 EXAMPLE 10

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MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. EXAMPLE 7 Verify a tangent to a circle SOLUTION Use the Converse of the Pythagorean Theorem. Because = 37 2, △ 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. 10. In the diagram, PT is a radius of ʘ P. Is ST tangent to ʘ P ?

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MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. 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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MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. RS is tangent to ʘ C at S and RT is tangent to ʘ C at T. Find the value of x. SOLUTION RS = RT 28 = 3x = x Substitute. Solve for x. Tangent segments from the same point are 11.

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MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. 12.

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MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. Homework Page 187 # 18 – 24 all Page 188 # 1 – 10.

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