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1 GPS UNIT 3 Semester 1 NLYTI GEOMETRY Page 1 of 3 ircles and Volumes Name: ate: Understand and apply theorems about circles M9-1.G..1 Prove that all circles are similar. M9-1.G.. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. M9-1.G..3 onstruct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. M9-1.G..4 (+) onstruct a tangent line from a point outside a given circle to the circle. Find arc lengths and areas of sectors of circlesm9-1.g..5 erive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of e angle as the constant of proportionality; derive the formula for the area of a sector. Explain volume formulas and use them to solve problems M9-1.G.GM.1 Give formal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, avalieri s principle, and informal limit arguments. M9-1.G.GM. (+) Give an informal argument using avalieri s principle for the formulas for the volume of a sphere and other solid figures. M9-1.G.GM.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Lesson 3.1 Properties of Tangents Goal: Use properties of a tangent to a circle. circle is the set of all points in a plane that are equidistant from the center of a circle. radius is a segment whose endpoints are the center and any point on the circle. chord is a segment whose endpoints are on the circle. diameter is a chord that contains the center of a circle. The diameter is twice the radius. secant is a line that intersects the circle in two points. tangent is a line in the plane of the circle that intersects the circle in one point only. Theorem 1: tangent line will be perpendicular to a radius of the circle at the point of tangency. Theorem : Tangent segments from common external points are congruent. PROLEMS ased on the definitions above draw the following lines. 1. Radius. hord 3. iameter 4. Secant 5. Tangent efine: P XY P Y X Note: P is the center of the circle

2 GPS UNIT 3 Semester 1 NLYTI GEOMETRY Page of 3 Example 1: raw common tangents. Tell how many common tangents the circles have by drawing them. a. b. Solution: a. b. You can fit 3 common tangents. You can fit 1 common tangent. PROLEMS: Tell how many common tangents the circles have by drawing them

3 GPS UNIT 3 Semester 1 NLYTI GEOMETRY Page 3 of 3 Example : Verify a tangent to a circle. Show that E is tangent to the circle. =17, E=8, =15 E Solution: To prove that E is tangent to the circle with radius E you must show that E is perpendicular to E (Theorem 1). Therefore, apply Pythagorean theorem: E E Since the left side equals the right side, we have a right triangle and E is a tangent. PROLEMS: Verify a tangent to a circle. Show that is tangent to the circle (or nota tangent to the circle). 4. =10, =6, =8 5. = 340, =1, =14 6. =14, =7, =9

4 GPS UNIT 3 Semester 1 NLYTI GEOMETRY Page 4 of 3 Example 3: is the point of tangency. Find the radius of the circle. r R 50 r 70 Solution: From Theorem 1 you know that is perpendicular to the tangent. Use the Pythagorean theorem to solve for the radius, r: ( r 50) r 70 ( r 50)( r 50) r 70 r 100r 500 r r 400 r 4 PROLEMS: is the point of tangency. Find the radius of the circle. 7. r r r r 14

5 GPS UNIT 3 Semester 1 NLYTI GEOMETRY Page 5 of 3 Example 4: RS is tangent to the circle at S and RT is tangent to the circle at T. Find the value of x. S 4 R PROLEMS: is tangent to the circle at and is tangent to the circle at. Find the value of x. 9. T x+4 Solution: RS RT Tangent segments from the same point are congruent. 4 x 4 Substitute. x 10 Solve for x. 5 3x x 16

6 GPS UNIT 3 Semester 1 NLYTI GEOMETRY Page 6 of 3 Lesson 3. Properties of ircles ircles and their Relationships among entral ngles, rcs, and hords In this unit you will study properties of circles. s you progress through the unit, you will learn new definitions and theorems. In some cases you will discover, or develop and prove these theorems. It may be advantageous for you to keep a ircle ook or list that includes the definitions and theorems addressed in each task. With each definition and theorem you enter, you should also include an illustrative sketch. We will begin by re-visiting the definition of a circle. Now we will introduce some notation and terminology needed to study circles. onsider the figure at right. m P = 75 ircles are identified by the notation P, where P represents the point that is the center of the circle. central angle of a circle is an angle whose vertex is at the center of the circle. P is a central angle of P. portion of a circle s circumference is called an arc. n arc is defined by two endpoints and the points on the circle between those two endpoints. If a circle is divided into two unequal arcs, the shorter arc is called the minor arc and the longer arc is called the major arc. If a circle is divided into two equal arcs, each arc is called a semicircle. diameter cuts a circle into two equal arcs. In our figure below, we call the portion of the circle between and including points and, arc notated by. We call the remaining portion of the circle arc, or. Note that major arcs are usually named using three letters. The central angle P has the same measure (in degrees) as the arc. Note that when we refer to the arc of a central angle, we usually mean the minor arc unless otherwise stated. PROLEMS: Fill in the missing terms. of a circle is an angle whose vertex is the center of the circle is an arc with endpoints that are the endpoints of a diameter. Using the picture below: n is portion of circumference also known as an unbroken part of a circle. If 180, then the points on the circle that lie on the interior of form a with endpoints and. This minor arc is named. The points on the circle that do not lie on the minor arc form a with endpoints and. The major arc is named.

7 GPS UNIT 3 Semester 1 NLYTI GEOMETRY Page 7 of 3 Example 1: Find the measure of each arc of the circle with center, diameter and 40. a.. b.. c.. Solution: a. is a semicircle and therefore 180. b. 40. That means that 40 as well. Therefore, c. is and PROLEMS: Find the measure of each arc. E is the diameter E E 4. E 5. E 6. E E 9. E

8 GPS UNIT 3 Semester 1 NLYTI GEOMETRY Page 8 of 3 Find the measure of each arc. E is a diameter. E F F 1. E 13. F 14. F 15. E Find the measure of each arc. E is a diameter E F 16. E 17. EF 18. E E 1. F

9 GPS UNIT 3 Semester 1 NLYTI GEOMETRY Page 9 of 3 Lesson 3.3 Properties of ords Goal: Use relationships between arcs and chords in a circle. Theorem 1: In the same circle two minor arcs are congruent if and only if their congruent chords are congruent. if and only if. Example: If 95, then. Theorem : If one chord is a perpendicular bisector of another chord, then the first chord is a diameter. If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc. E If is to, then E E and. Example: If If is to and E 6, then E 6 and. Theorem 3: In the same circle two chords are congruent if and only if they are equidistant from the center. M if and onlyif KL KM. L K Example: If KL KM and 8, then 8. PROLEMS: 1. If 15, find.. If 105, find. 3. If 0 and 60, find. 4. If 0 and 95, find. 5. If 5 and 90, find.

10 GPS UNIT 3 Semester 1 NLYTI GEOMETRY Page 10 of 3 Example : 1. MO 3x and ON 7x 8. Find MO. N M O Solution: hords and are congruent, therefore they are equidistant from the center, O. That means MO ON. Problems: In the diagram in example, suppose 16 and MO Find MO ON y theorem 3. 3x 7x 8 Substitute. x Solve for x. So MO 3x 3() Find M 6. Find ON In the diagram in example, suppose 4, MO x and ON 4x Find 8. Find x 9. Find ON

11 GPS UNIT 3 Semester 1 NLYTI GEOMETRY Page 11 of 3 Find the measure of the given arc or chord Find x. 1x+7 3x+16

12 GPS UNIT 3 Semester 1 NLYTI GEOMETRY Page 1 of 3 Lesson 3.4 Inscribed ngles and Polygons Goal: Use inscribed angle of circles. inscribed angle is an angle whose vertex is on a circle and whose sides contain chords of a circle. Inscribed ngle Theorem 1: The measure of an inscribed angle is one half the measure of its intercepted arc. Example: If 40 then Theorem : If two inscribed angles of a circle intercept the same arc, then the angles are congruent. Example: If 100 then 50 Theorem 3: If a right angle is inscribed in a circle, then the hypotenuse is a diameter of a circle. Example: Since the right angle is inscribed in the circle, the hypotenuse is the diameter,. Theorem 4: quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary. Example: y 180 x x y

13 GPS UNIT 3 Semester 1 NLYTI GEOMETRY Page 13 of 3 PROLEMS: Use Theorems 1 and to solve the problems. 9. What is? What is? What is E? 41 E 1. If NO 70, what is NPO? M N 13. If MP 106, what is MOP? P O 14. Which two pairs of angles are congruent? 15. Find the measure of the indicated angle or arc: a. b. 63 P P 115 E Find. Find E.

14 GPS UNIT 3 Semester 1 NLYTI GEOMETRY Page 14 of 3 Use Theorem 4 to solve the problems. 16. Find x and y y 4x 17. Find x and y. x 8y y +80 x 18. Find x and y. 5x y 8y x

15 GPS UNIT 3 Semester 1 NLYTI GEOMETRY Page 15 of 3 Lesson 3.5 Other ngle Relationships in ircles Goal: Find the measures of angles inside or outside a circle. Theorem 1:If a tangent and a chord intersect at a point on the circle, then 0 1 the measure of each angle formed is one half the measure of its intercepted arc. Example: If 0 then Theorem : If two chords intersect inside a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle. Example: If 86 and 78 then x x x 78 Theorem 3: If a tangent and a secant, two tangents, or two secants intersect outside a circle, then the measure of the angle formed is one-half the difference of the measures of the intercepted arcs. Example: If 15 and 5, then x 45 x PROLEMS: Use Theorem 1 to solve the problems. 170 x 1. What is the measure of x?. What is the measure of x? x 70

16 GPS UNIT 3 Semester 1 NLYTI GEOMETRY Page 16 of 3 Use Theorem to solve the problems What is the measure of x? x 85 x 4. What is the measure of x? What is the measure of x? x 6. What is the measure of x? x 98 74

17 GPS UNIT 3 Semester 1 NLYTI GEOMETRY Page 17 of 3 Use Theorem 3 to solve the problems. 7. What is the measure of x? 15 x 5 8. What is the measure of x? 185 x What is the measure of x? 85 x

18 GPS UNIT 3 Semester 1 NLYTI GEOMETRY Page 18 of 3 Lesson 3.6 Segment Lengths in ircles Goal: Find segment lengths in circles. Theorem 1: If two chords intersect in the interior of a circle, then the products of the lengths of the segments of one chord is equal to the products of the lengths of the other chord. O In other words: O O O O Theorem : If two secant segments share the same endpoint outside a circle, then the product of the lengths of one entire secant segment and its external segment equals the product of the lengths of the other entire secant segment and its external segment. In other words: E E E E E Theorem 3: If a secant segment and a tangent segment share an endpoint outside a circle, then the product of the lengths of the entire secant segment and its external segment equals the square of the length of the tangent segment. In other words: E E E E Example 1: Find O. Solution: Use Theorem O x 10 O O O O 6 x 8 10 x 13.3

19 GPS UNIT 3 Semester 1 NLYTI GEOMETRY Page 19 of 3 PROLEMS: Use Theorem 1 to solve the problems. 1. What is the measure of x? 8 1 x 6. What is the measure of x? x 3. What is the measure of x? 0 16 x 16

20 GPS UNIT 3 Semester 1 NLYTI GEOMETRY Page 0 of 3 Example : Find E Solution: Use Theorem. E E E E x ( x 9) 4 (4 5) x 9 x x 9x 36 E 4 5 9x 36 0 ( x 1)( x 3) 0 x 1 and x 3 u substitution: Length cannot be negative, so throw out x = -1. E = x + 9 =3 + 9 = 1 Use Theorem to solve the problems. 4. What is the measure of x? x What is the measure of x? x 4 8

21 GPS UNIT 3 Semester 1 NLYTI GEOMETRY Page 1 of 3 Example 3: Find E. Solution: Use Theorem 3. E E E 6 6 x ( x 9) x E x 9 36 x 9 9x 36 0 ( x 1)( x 3) 0 x 1 and x 3 Length cannot be negative, so E = x = 3. x 6. What is the measure of x? x What is the measure of x? 4 9 x

22 GPS UNIT 3 Semester 1 NLYTI GEOMETRY Page of 3 Lesson 3.7 ircumference and rc Length Goal: Find arc length and other measures of circles. The circumference of a circle is the distance around the circle. n arc length is a portion of the circumference of a circle. Theorem 1: The circumference of a circle is ror d, where r = radius and d = diameter. In a circle the ratio of the length of a given arc to the circumference is equal to the ratio of the measure of the arc to 360. Example 1: a. Find the circumference of a circle with a radius of 5 meter. Solution: r meter. b. Find the diameter of a circle with a circumference of 0 meters. Solution: d 0 d 0 d 6.4 meters PROLEMS: Find the indicated measure. 1. The circumference of a circle with a radius of 1 meters.. The radius of a circle with a circumference of 5 feet. 3. The diameter of a circle with a circumference of 50 meters. 4. The radius of a circle with a diameter of 10 feet.

23 GPS UNIT 3 Semester 1 NLYTI GEOMETRY Page 3 of 3 Example : Find the length of. Radius = 5 meters. 130 Solution: meters. 360 The ratio of the arc to the circumference is the same as the ratio of the angle to 360 Solve for. PROLEMS: Find the length of ft 7m 1 cm

24 GPS UNIT 3 Semester 1 NLYTI GEOMETRY Page 4 of 3 Example 3: Find the circumference of E. 60 E 81.6 cm Solution: 60 ircumference 360 The ratio of the arc (81.6 cm) to the circumference is the same as the ratio of the angle to ircumference cm. 60 Solve for the circumference. PROLEMS: Find the indicated measure. 8. m 9. Radiusof E. 10. ircumferenceof E. T 9 in 45 cm ft E 55 m E S

25 GPS UNIT 3 Semester 1 NLYTI GEOMETRY Page 5 of 3 Find the perimeter of the region cm 0 cm r=10 cm 1. Radius = 3 cm 0 cm 7 cm 13. radius = 5 cm 15 cm

26 GPS UNIT 3 Semester 1 NLYTI GEOMETRY Page 6 of 3 Lesson 3.8 reas of ircle and Sectors Goal: Find the areas of circles and sectors. sector of a circle is the region bounded by two radii of the circle and their intercepted arc. Theorem 1: The area of a circle is times the square of the radius. ( r ) Theorem : The ratio of an area of a sector of a circle to the area of the whole circle is equal to the ratio of the measure of the intercepted arc to 360. Example 1: Find the indicated measure. a. Radius b. rea r 5 in rea= 0 m² Solution: r 0 r r 5 0 r PROLEMS: Find the indicated measure. 1. The diameter of a circle is 1 centimeters. Find the area.. The area of a circle is square meters. Find the radius. 3. The area of a circle is 500π square meters. Find the diameter.

27 GPS UNIT 3 Semester 1 NLYTI GEOMETRY Page 7 of 3 Example : Find the area of the lined sector. 11 in 165 Solution: 165 Shaded rea 360 Entire rea of ircle 165 Shaded rea y Theorem Substitute 165 Shaded rea in Solve 360 PROLEMS: Find the area of the lined sector cm in cm 75

28 GPS UNIT 3 Semester 1 NLYTI GEOMETRY Page 8 of 3 Lesson 3.9 Surface reas and Volumes of Spheres Goal: Find the surface areas and volumes of spheres. sphere is the set of all points in space which are equidistant from a given point (= center of sphere). typical example of a sphere is a round ball. Theorem 1: The surface area, S, of a sphere is: S 4 r Theorem : The volume, V, of a sphere is: V 4 r 3 3 Example 1: Find surface area and volume of a sphere with a radius of 8 inches. Solution: Surface rea S 4 r in Volume 4 3 V r in 3 PROLEMS: 1. Find surface area and volume of a sphere with a radius of 1 cm.. Find surface area and volume of a sphere with a diameter of 15 cm. 3. Find radius of a sphere with a surface area of 315 m². 4. Find the diameter of a sphere with a volume of 10 cubic inches.

29 GPS UNIT 3 Semester 1 NLYTI GEOMETRY Page 9 of 3 Lesson 3.10 avalieri and Volumes of ylinders, Pyramids, and ones avalieri s principle: If, in two solids of equal altitude, the sections made by planes parallel to and at the same distance from their respective bases are always equal, then the volumes of the two solids are equal (Kern and land 1948, p. 6). oth coin stacks have the same volume since the sections made by planes parallel to and at the same distance from their respective bases are always equal. You could also say they have the same amount of identical coins ( cent Euro coins). That is pretty much avalieri s principle in simple terms. Problems: t any given height, the horizontal length of the figure on the left is 1 cm. In problems 1 5 you will calculate the figure s area. Now use 10 congruent rectangles to approximate the figure (remember l = 1cm) 1. What is the length of each rectangle?. What is the height of each rectangle? 3. What is the area of each rectangle? 4. What is the area of 10 rectangles?

30 GPS UNIT 3 Semester 1 NLYTI GEOMETRY Page 30 of 3 5. Suppose you use 100 rectangles a. What is the area of each rectangle in that case? b. What is the area of all rectangles in that case? 6. s n gets larger, do the rectangles approximate the figure better, worse, or the same? Example 1: Find volume of a cylinder radius of inches and a height of 6 inches. Solution: The formula for the volume of a cylinder is V base height wherebase r cylinder. Plug the numbers into formula and solve for the unknown variable: V base height r height in cylinder 3 Problems: 7. Find volume of a cylinder radius of 5 cm and a height of 3 cm. 8. What is the height of a cylinder with a volume of 50 cm³ and radius of 3 cm? 9. What is the radius of a cylinder with a volume of 3 m³ and a height of m?

31 GPS UNIT 3 Semester 1 NLYTI GEOMETRY Page 31 of 3 Example : Find volume of a pyramid with a square base of inches and a height of 6 inches. Solution: The formula for the volume of a pyramid is Vpyramid 1 3 b h. Plug the numbers into formula and solve for the unknown variable: V base height r height in cylinder 3 Problems: 10. Find volume of a pyramid with a base length of 5 cm and a height of 0 cm. 11. What is the height of a pyramid with a volume of 50 cm³ and base length of 3 cm? 1. What is the base area of a pyramid with a volume of 3 m³ and a height of 1 m?

32 GPS UNIT 3 Semester 1 NLYTI GEOMETRY Page 3 of 3 Example 3: Find volume of a cone with a radius of inches and a height of 6 inches. Solution: The formula for the volume of a cone is Vcone 1 3 r h. Plug the numbers into formula and solve for the unknown variable: 1 1 Vcone r h 6 5.1in Problems: 13. What is the volume of an ice cream cone (which you can buy from Ms. onnell) with a radius of 1 inch and a height of 3 inches? 14. What is the radius of a cone with a volume of 3 m³ and a height of m? 15. What is the height of a cone with a volume of 100 m³ and a radius of m?

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