Geometry P Lesson 10-1: ircles and ircumference Page 1 of 1 Objectives: To identify and use parts of circles To solve problems involving the circumference of a circle Geometry Standard: 8 Examples: The name of the circle is: The radii of the circle are: The chords of the circle are: The diameter of the circle is: If = 18 mm, then R = If RY = 10 in, then R = and = Is XY? Explain Find the exact circumference of the circle with the given information in terms of. 5. d = 12 mm 6. r = 3 5 yd 7. 8.
Geometry P Lesson 10-2: Measuring ngles and rcs Page 1 of 2 Objectives: Recognize major arcs, minor arcs, semicircles, and central angles and their measures Find arc length Geometry Standard: 7 central angle is an angle whose vertex is at the of a circle and whose sides are. central angle separates a circle into two arcs, a major arc and a minor arc. HE is a central angle. Name 4 other central angles: The sum of the measures of the central angles (that don t overlap) of a circle is. If m HE 155 m EF 85 m HEG m GEF, find m HEG minor arc has a measure less than 180. Use 2 letters to name a minor arc. o 3 minor arcs are: RS,, minor arc has the same measure as its central angle. o mtu = 60 Find msr and msq major arc has a measure greater than 180. Use 3 letters to name a major arc. o 3 major arcs are: RTQ,, The measure of a major arc is 360 minus the measure of the minor arc with the same endpoints. o Find mqts = 360 90 = Find mtqu semicircle is an arc that measures 180. Use 3 letters to name a semicircle. o Name a semicircle: o The endpoints of a semicircle are the endpoints of a of a circle. Two arcs are congruent if and only if their central angles are congruent N they are in the same circle or in congruent circles (circles with the same radius). 85 R M 85 G O 85 U T Which arcs have the same measure? Which arcs are congruent? djacent arcs have exactly one point in common. djacent arcs can be added. o Name 2 adjacent arcs: rc ddition Postulate: m m m 44 m O m = m = m =
Geometry P Lesson 10-2: Measuring ngles and rcs Page 2 of 2 rc Length The measure of an arc is different than the length of an arc. The measure of an arc is in degrees. The length of an arc is in units such as cm, mm, inches, etc. n arc is part of a circle and its length is a portion of the circumference of the circle. Use this proportion to find the length of an arc: Example: OE = 12 mm, m OE = 120, m O = 45 me = length E (in terms of ) m = length (in terms of ) Example: MT = 15 cm, SPT RPT If m SPT 70, find length of RT If m RPT 50, find length of NR Find length of MST Summarize what you learned in today s lesson.
Geometry P Lesson 10-3: rcs and hords Page 1 of 2 Objectives: Recognize and use relationships between arcs and chords. Recognize and use relationships between chords and diameters. Geometry Standards: 7, 21 The endpoints of a chord are also the endpoints of an arc. In this lesson, you will learn three theorems involving chords and arcs. Theorem 10.2 In a circle or in congruent circles, two minor arcs are congruent if and only if their corresponding are congruent. so which arcs congruent? m O = 50 so what are the measures of,,? Find m Find mp Theorem 10.3 In a circle, if a diameter (or radius) is perpendicular to a chord, then it. O K OK so which segments congruent? Which arcs are congruent? If OE, O = 15, and = 24, find x. Theorem 10.4 - In a circle or in congruent circles, two chords are congruent if and only if they are equidistant from the of the circle. E G G and F are equidistant to E so which chords are congruent? F
Geometry P Lesson 10-3: rcs and hords Page 2 of 2 If all the vertices of a polygon lie on a circle, then the polygon is said to be inscribed in the circle and the circle is circumscribed around the polygon. RSTV is inscribed in circle O. ircle O is circumscribed around RSTV. raw a right triangle inscribed in a circle. regular pentagon is inscribed in a circle. Find the measure of each arc of the circle. circle is circumscribed around the isosceles triangle below. Solve for x, muv, mut, and mtv raw a circle with center W. raw diameter ML raw a chord HK which is perpendicular to ML, but does not pass through W. Label the intersection of ML and HK point J. Let WL = 10 cm and HK = 16 cm. Use trigonometry to find m HWJ Find mhl Find mmk Find JL
Geometry P Lesson 10-4: Inscribed ngles Page 1 of 2 Objective: Find measures of inscribed angles and polygons Geometry Standards: 7, 21 n inscribed angle is an angle whose vertex is on the circle and whose sides are of the circle. N G inscribed angle NG central angle NG Inscribed ngle Theorem If an angle is inscribed in a circle, then the measure of its arc equals the measure of the angle. O x N G If mg = 140, then m OG = If m OG = 80, then mg = 2x O G 65 N J What is the intercepted arc for JON? Name all angles that intercept O Find mjon P m = 100 m = m P = Theorem 10.6 If two inscribed angles intercept the same arc or congruent arcs, then the angles are. Which angles in the figure above (left) are congruent because of Theorem 10.6? Theorem 10.7 If an inscribed angle intercepts a semicircle, then it is a. intercepts semicircle therefore m = Name another right angle and its semicircle Theorem 10.8 If a quadrilateral is inscribed in a circle, then its opposite angles are. T S O H T and H are opposite angles therefore they are. Name another pair of supplementary angles
Geometry P Lesson 10-4: Inscribed ngles Page 2 of 2 E H T Y HEY is inscribed in. EH EY Find x and EH, HY = 12, m E = 3x U S 60 X T US is inscribed in X. What is m S? If US = 9, then find U and S Probability You can also use the measure of an inscribed angle to determine the probability of a point lying on an arc. 60 Points and T are on a circle so that mt 60. Suppose point is T randomly placed on the circle (not at or T). What is the probability that m T 30? Suppose point is randomly placed on V (not at R or S). What is the probability that m RS 70?
Geometry P Lesson 10-5: Tangents Page 1 of 2 Objectives: Use properties of tangents. Solve problems involving circumscribed polygons. Geometry Standards: 7, 21 tangent to a circle intersects the circle in exactly one point, called the. In this lesson, you will learn three important theorems involving tangents. V T R O P S VW is tangent to O at point. Name another tangent and its point of tangency. W U Theorem 10.9 If a line is tangent to a circle, then it is to the radius drawn to the point of tangency. R T P U S RS is tangent to P. Therefore RS RP. What other segments are? Theorem 10.10 If a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is to the circle. EF E, therefore EF is tangent to. Name any other tangents: Find x Theorem 10.11 If two segments from the same exterior point are tangent to a circle, then the segments are. and are tangent to, therefore. These other items are congruent too. Explain why.
Geometry P Lesson 10-5: Tangents Page 2 of 2 Solve for x Solve for x Solve for x and y 15 5 x 30 y Solve for x 3x 4 x 2 x 8 When a polygon is circumscribed around a circle, all of the sides of the polygon are to the circle. ll sides of are tangent to O. What does Theorem 10.11 say about and E? Find,, E and the perimeter of. Solve for x Solve for x Solve for x and y y
Geometry P Lesson 10-6: Secants, Tangents, and ngle Measures Page 1 of 2 Objective: Find measures of angles formed by lines intersecting on, inside or outside a circle. Geometry Standards: 7, 21 line that intersects a circle in exactly two points is called a secant. X Name a secant: Name a tangent: Name 2 chords: E Name a radius: Theorem 10.12 If 2 secants (or 2 chords) intersect in the interior of a circle, then the measure of the angle formed is. m 1 m X 120 X 40 Theorem 10.13 If a secant (or chord) and a tangent intersect at the point of tangency, then the measure of each angle formed is. m XTV m 6 m YTV Theorem 10.14 If a tangent and a secant, 2 tangents, or 2 secants intersect in the exterior of a circle, then the measure of the angle formed is of the intercepted arcs. X P W U K S Z R Q Y m K = m U = m S =
Geometry P Lesson 10-6: Secants, Tangents, and ngle Measures Page 2 of 2 Important: First determine where the vertex of the angle is. This will help you figure out what formula to use. Vertex is inside the circle: entral angle: m = measure of intercepted arc 2 secants (or 2 chords): m = ½ the sum of intercepted arcs Vertex is on the circle: Inscribed angle: m = ½ measure of intercepted arc Tangent and a secant (or chord): m = ½ measure of intercepted arc Vertex is outside the circle: Two tangents: Two secants: m = ½ the difference of intercepted arcs Secant & tangent:
Geometry P Lesson 10-7: Special Segments in a ircle Page 1 of 2 Objectives: Find measures of segments that intersect in the interior or the exterior of a circle Geometry Standards: 7, 21 We ve studied numerous theorems in this chapter on angle and arc measures. Now we will switch to segment lengths. Segments that intersect inside a circle Theorem 10.15 If two chords intersect inside a circle, then the product of the segments of one chord is equal to the product of the segments of the other chord. E Segments that intersect outside a circle Theorem 10.16 If 2 secant segments share the same endpoint outside the circle, then the product of the measures of one secant segment and its external segment equals the product of the measures of the other secant segment and its external segment. Solve for k: Theorem 10.17 If a secant segment and a tangent segment are drawn to a circle from the same exterior point, then the square of the measure of the tangent segment is equal to the product of the measures of the secant segment and its external segment. E and are chords. E E and E are secant segments. E and E are external segments. E is a tangent segment. E is a secant segment. E is an external segment. 18 9 x 12 10 11 10 k Solve for x: 12 Solve for x: 12 x
Geometry P Lesson 10-7: Special Segments in a ircle Page 2 of 2 Examples: Solve for x.
Geometry P Lesson 10-8: Equations of ircles Page 1 of 2 Objectives: Write the equation of a circle Graph a circle on the coordinate plane Geometry Standards: 7, 21 We ve studied equations of lines, what about equations of circles? efinition of a circle: The locus of points in a plane equidistant from a given point. The general equation of a circle is: Where the center is located at and the radius is Note: The equation of a circle is kept in the form above. The terms being squared are not expanded. Example 1: 1 y 7 25 Example 2: x y 2 2 x enter is at: Radius is: 2 2 4 6 7 enter is at: Radius is: 2 2 Example 3: x y 16 enter is at: Radius is: Example 4: enter (0, 3) radius = 4 Example 5: enter (-5, 8) radius = 10 Equation: Equation: Example 6: enter (6, -1), Pt. on the circle: P(1, 2) Equation: Example 7: Endpoints of the diameter are (-1, 3) (-5, 7) Equation: Example 8: What is the exact circumference of the circle with equation x 1 y 7 25 2 2
Geometry P Lesson 10-8: Equations of ircles Page 2 of 2 2 2 2 Example 9: Find the radius of the circle that has equation 5 3 (5, 1). x y r and passes through Graphing ircles o Step 1: Locate the center of the circle o Step 2: Using the radius, plot 4 points from the center of the circle. o Step 3: onnect the 4 points with arcs that each represent ¼ of the circle. 2 2 Example 10: Graph x 2 y 1 9 Example 11: Graph x 2 y 16 1 2-10 -10-10 1-10 1 0 0 2 2 Example 12: Graph x y 4 Example 13: Equation -10-10 -10 1-10 1 0 0