Presentation on theme: "Arcs Tangents Angles Sphere Circumference What is Unit 3 about?"— Presentation transcript:
1 ArcsTangentsAnglesSphereCircumferenceWhat 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.
2 Crop CirclesWhether 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!
3 Properties of Tangents Tuesday, April 11, 2017Essential Question:How do we identify segments and lines related to circles and how do we use properties of a tangent to a circle?Lesson 6.1M2 Unit 3: Day 1
4 Warm Ups1. What measure is needed to find the circumference or area of a circle?ANSWERradius or diameter2. Find the radius of a circle with diameter 8 centimeters.ANSWER4 cm3. A right triangle has legs with lengths 5 inches and 12 inches. Find the length of the hypotenuse.ANSWER13 in.4. Solve 6x + 15 = 33.5. Solve (x + 18)2 = xANSWER7ANSWER3
6 CircleThe set of all points in a plane that are equidistant from a given point, called the center.EFCircleTangentChordPDiameterRadiusDASecantCBName of the circle: ʘ P
7 DefinitionRadius – a segment from the center of the circle to any point on the circle.Diameter – a chord that contains the center of the circle.CAB
8 DefinitionChord – a segment whose endpoints are points on the circle.
9 DefinitionSecant – a line that intersects a circle in two points.
10 DefinitionTangent – a line in the plane of a circle that intersects the circle in exactly one point.PPoint of tangencyO
11 A little extra information The word tangent comes from the Latin word meaning to touchThe word secant comes from the Latin word meaning to cut.
12 EXAMPLE 11. Tell whether the line or segment is best described as a chord, a secant, a tangent, a diameter, or a radius.tangentdiameterchordradius
13 Identify special segments and lines EXAMPLE 2Identify special segments and lines2. Tell whether the line, ray, or segment is best described as a radius, chord, diameter, secant, or tangent of ʘ C.ACa.b.ABDEc.AEd.SOLUTIONis a radius because C is the center and A is a point on the circle.ACa.b.ABis a diameter because it is a chord that contains the center C.c.DEis a tangent ray because it is contained in a line that intersects the circle at only one point.d.AEis a secant because it is a line that intersects the circle in two points.
14 EXAMPLE 3Find lengths in circles in a coordinate plane3. Use the diagram to find the given lengths.a.Radius of ʘ Ab.Diameter of ʘ ARadius of ʘ Bc.Diameter of ʘ Bd.SOLUTIONa.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.
15 GUIDED PRACTICEEXAMPLE 44. Find the radius and diameter of ʘ C and ʘ D.SOLUTIONa.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.
16 DefinitionsCommon tangent – a line or segment that is tangent to two coplanar circlesCommon internal tangent – intersects the segment that joins the centers of the two circlesCommon external tangent – does not intersect the segment that joins the centers of the two circles
17 5. Tell whether the common tangents are internal or external. EXAMPLE 55. Tell whether the common tangents are internal or external.a.b.common internal tangentscommon external tangents
18 Draw common tangentsEXAMPLE 66. Tell how many common tangents the circles have and draw them.a.b.c.SOLUTIONa.4 common tangents3 common tangentsb.c.2 common tangents
19 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.
20 Tangent TheoremsCreate right triangles for problem solving.
21 Write the binomial twice. Multiply. EXAMPLE 8Find the radius of a circle7. In the diagram, B is a point of tangency. Find the radius r of ʘC.SOLUTIONYou know that AB BC , so △ ABC is a right triangle. You can use the Pythagorean Theorem.AC2 = BC2 + AB2Pythagorean Theorem(r + 50)2 = rSubstitute.(r + 50)(r + 50) = rWrite the binomial twice.Multiply.r2 + 50r +50r = rr r = rCombine Like Terms.100r = 3900Subtract from each side.r = 39 ft .Divide each side by 100.
22 Subtract from each side. EXAMPLE 88. ST is tangent to ʘ Q. Find the value of r.SOLUTIONYou know from Theorem 10.1 that ST QS , so △ QST is a right triangle. You can use the Pythagorean Theorem.QT2 = QS2 + ST2Pythagorean Theorem(r + 18)2 = rSubstitute.r2 + 36r = rMultiply.36r = 252Subtract from each side.r = 7Divide each side by 36.
23 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.
24 GUIDED PRACTICEEXAMPLE 109. Is DE tangent to ʘ C?ANSWERYes – 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
25 Verify a tangent to a circle EXAMPLE 710. In the diagram, PT is a radius of ʘ P. Is ST tangent to ʘ P ?SOLUTIONUse 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.
26 Congruent Tangent Segments Theorem 6.2 If two segments from the same exterior point are tangent to a circle, then they are congruent.
27 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.SOLUTIONRS = RTTangent segments from the same point are28 = 3x + 4Substitute.8 = xSolve for x.