Circle: The set of points in a plane that are fixed distance from a given point called the center of the circle. Unit 4 Reference Sheet Chord: A segment whose endpoints both lie on the same circle. Name: Radius: A segment whose endpoints are the center of a circle and a point on the circle. chord Diameter: A segment that has endpoints on a circle and that passes through the center of the circle. Secant: A line that intersects a circle at two points. Tangent: A line that is in the same plane as a circle and intersects the circle at exactly one point. The radius is perpendicular to the tangent at the point of tangency. Internal Tangent: A tangent that is common to two circles and intersects the segment joining the centers of the circles. External Tangent: A tangent that is common to two circles and does not intersect the segment joining the centers of the circles. Inscribed Angle: An angle whose vertex is on a circle and whose sides contain chords of the circle. The angle measure of the inscribed angle is ½ of the intercepted arc. Central Angle: An angle whose vertex is the center of a circle. Angle Between 2 Chords: The angle between 2 chords is equal to ½ the sum of the two intercepted arcs. Angle Between 2 Secants: The angle between 2 chords is equal to ½ the difference of the two intercepted arcs. M. Winking (Unit 4-00) p.87
1. Sec 4.1 Circles & Volume The Language of Circles Using the Pythagorean Theorem to find the value of x in each of the diagrams below: 1. 2. Name: Converse of the Pythagorean Theorem. Which of the following are right triangles? 11 3. 4. 5. 6. 7. 15 17 8 35 37 8 17 15 12 8 5 6 4 Right Triangle? (circle one) YES NO Right Triangle? (circle one) YES NO Right Triangle? (circle one) YES NO Right Triangle? (circle one) YES NO Right Triangle? (circle one) YES NO 8. G A. Diameter 9. A B. Radius 10. DE C. Center 11. GC D. Secant 12. JB E. Chord 13. HJ F. Point of tangency 14. HI G. Common external tangent 15. AB H. Common internal tangent Determine if AB is tangent to the circle centered at point C. Explain your reasoning. 16. 17. M. Winking Unit 4-1 page 88
AB and AD are tangent to the circle centered at point C. Find the value of x. 18. 19. Given the center of the circle is point A, find the requested measure. 20. mef = 21. mce = 22. mcdf = 23. mde = 24. mbc = 25. mfb = 26. mfbe = 27. mdfc = 28. mdfb = 29. mbec = Determine the measure of BC. 30. 31. 32. Find the requested measure for each circle. 33. FC = 34. mbg M. Winking Unit 4-1 page 89
1. Sec 4.2 Circles & Volume Inscribed Angles Name: Central Angle: An angle whose vertex is the center of the circle. Inscribed Angle: An angle whose vertex is on a circle and whose sides contain chords of the circle Central Angle Inscribed Angle Inscribed Angle Properties: Consider the following diagram an inscribed angle of the circle center at A. D D D D C A B C A B C A B C A B Consider the inscribed angle CBD which intercepts arc DC that measures 70. Since the central angle CAD intercepts arc DC then m CAD = 70. Triangle DAB is isosceles because the legs are radii of the circle. The measure of angle m DAB = 110 since it forms a linear pair with CAD. The based angles of DAB must be congruent and the interior angles of triangle must sum to 180. So, 110 + x + 180 In a similar fashion using addition or subtraction, it can be shown this idea extends to any inscribed angle. An inscribed angle s measure is exactly half of the arc measure that it intercepts. Find the most appropriate value for x in each of the diagrams below. (Assume point A is the center of the circle.) 1. 2. 3. A A A M. Winking Unit 4-2 page 90
Find the most appropriate value for x in each of the diagrams below. (Assume point A is the center of the circle.) 4. 5. 6. A A A 7. 8. 9. A A A 10. 11. 12. A A A M. Winking Unit 4-2 page 91
Find the most appropriate value for x in each of the diagrams below. (Assume point A is the center of the circle.) 13. 14. 15. A A A 16. 17. 18. A A A 19. 20. 21. A A A M. Winking Unit 4-2 page 92
1. Sec 4.2a Circles & Volume Tangent Circle Construction Name: [Creating a Tangent To a Circle] Construct a line tangent to circle with center A and passing through point C. Step I: First draw a segment with end points A & C. Step 2: Create a perpendicular bisector to segment AC Step 3: Create a circle centered at the midpoint of segment AC and with a radius from the midpoint to point A. Step 4: Draw a line that passes through point C and either of the intersections of the original circle and the newly created circle (point E in the diagram). Construct a tangent line to circle with center A that passes through point C. C A M. Winking Unit 4-a2 page 93
1. Sec 4.3 Circles & Volume Angles of Circles Name: Tangent Line Angles Consider the tangent line DC and the ray CB which intercepts arc BC and has a measure of x. Draw an auxiliary segment AB and AC to create an isosceles triangles. We know that m A = x as a central angle and the interior angles of ABC sum to 180. So, m B + m C = 180 x Also, m B = m C because they are the base angles of an isosceles triangle. So, m B + m B = 180 x which simplifies: 2 m B = 180 x or m B = 180 x 2 Finally, AC must be perpendicular to DC since it is tangent of the circle. The angle DCB & ACB must sum to 90. So, we can find 180 x m DCB = 90 2 which simplifies: m DCB = x 2 The measure of an angle formed by a tangent and a chord drawn to the point of tangency is exactly ½ the measure of the intercepted arc. Find the most appropriate value for x in each of the diagrams below. (Assume CE is tangent to the circle.) 1. 2. 3. M. Winking Unit 4-3 page 94
Find the most appropriate value for x in each of the diagrams below. (Assume CE is tangent to the circle.) 4. 5. 6. (You may assume DF is a diameter.) Intersecting Chords Interior Angles Consider the intersecting chords BE and FC that intercept the arcs CB and FE. Draw an auxiliary segment BF to create the inscribed angles that we know are half of the intercepted arc. So, m FBE = z y and m BFC = 2 2 Since triangles interior angles sum to 180. So we can subtract the 2 angles of triangle DBF to find angle m BDF = 180 y z 2 2 Finally, since FDE forms a linear pair with BDF we can subtract from 180 to find: m FDE = 180 (180 y z ) 2 2 which simplifies to m FDE = y +z 2 The measure of an angle formed by two intersecting chords of the same circle is exactly ½ the measure of the sum of the two intercepted arcs. Find the most appropriate value for x in each of the diagrams below. (Assume point A is the center of the circle.) 7. 8. M. Winking Unit 4-3 page 95
Find the most appropriate value for x in each of the diagrams below. (Assume point A is the center of the circle.) 9. 10. (You may assume BE is a diameter.) 11. 12. 13. 14. M. Winking Unit 4-3 page 96
Secant Lines Exterior Angle Consider the rays BD and BF which intercepts arc CE and FD which measure a and b respectively. Draw an auxiliary segment. ED We know that m DEF = b and 2 m CDE = a because each is an 2 inscribed angle. Finally, m DEB = 180 b since the two angles at point E forma linear pair. Furthermore, m B = 180 (180 b ) a 2 2 since a triangle s interior angles sum to 180 and that would simplify to m B = (b a). 2 2 The measure of an angle formed on the exterior of a circle by two intersecting secants of the same circle is exactly ½ the measure of the difference of the two intercepted arcs. Find the most appropriate value for x in each of the diagrams below. (Assume point A is the center of the circle.) 15. 16. 17. 18. M. Winking Unit 4-3 page 97
Find the most appropriate value for x in each of the diagrams below. (Assume point A is the center of the circle.) 19. 20. You may assume DB is tangent to the circle. 21. 22. You may assume EC and DE are tangent to the circle. You may assume DB is tangent to the circle. 23. 24. M. Winking Unit 4-3 page 98
1. Sec 4.4 Circles & Volume Circle Segments Name: Intersecting Chords Consider the intersecting chords DC and EF that intersect at point B. Draw an auxiliary segment DE and CF to create triangles DBE and FBC. We know that DEB FCB because they are both inscribed angles that intercept the same arc FD. Similarly, we know EDB CFB. Then, by AA we know DBE~ FBC Using proportions of similar triangles: x 1 y 2 = y 1 x 2 We can cross-multiply to give us the following statement: x 1 x 2 = y 1 y 2 Part1 Part2 Part1 Part2 If two chords intersect then the product of the measures of the two subdivided parts of one chord are equal to the product of the parts of the other chord. Find the most appropriate value for x in each of the diagrams below. 1. 2. 3. 4. M. Winking Unit 4-4 page 99
Find the most appropriate value for x in each of the diagrams below 5. 6. Segments of Secants Consider the intersecting segments of secants EC and AC that intersect at point C. Draw an auxiliary segment AD and BE to create triangles ADC and EBC. We know that CAD CEB because they are both inscribed angles that intercept the same arc BD. Reflexively, we also know C C. Then, by AA we know ADC~ EBC Using proportions of similar triangles: x 2 + x 1 y 1 + y 2 = y 1 x 1 We can cross-multiply to give us the following statement: (x 2 + x 1 ) x 1 = (y 1 + y 2 ) y 1 Whole External Whole External If 2 secants intersect the same circle on the exterior of the circle then the product of the whole and the external segment measures is equal to the same product of the other secant s portions. Find the most appropriate value for x in each of the diagrams below. 7. 8. M. Winking Unit 4-4 page 100
Find the most appropriate value for x in each of the diagrams below. 9. 10. 11. 12. Segments of Secants and Tangents Consider the intersecting segment of a secant AC and segment of a tangent AF that intersect at point A. Draw an auxiliary segment BD and CD to create triangles ADC and ABD. We know that BDA ACD because they are both have a measure of half of the intercepted arc BD. Reflexively, we also know A A. Then, by AA we know ADC~ ABD Using proportions of similar triangles: x 1 y 1 + y 2 = y 1 x 1 We can cross-multiply to give us the following statement: (y 1 + y 2 ) y 1 = x 1 x 1 Whole External Tangent Tangent M. Winking Unit 4-4 page 101
Find the most appropriate value for x in each of the diagrams below. 13. 14. a b = c d (a + b) a = (c + d) c (a + b) a = c 2 Part1 Part2 Part1 Part2 Whole External Whole External Whole External Tangent 2 Find the most appropriate value for x in each of the diagrams below. 13. 14. M. Winking Unit 4-4 page 102
1. Sec 4.5 Circles & Volume Circumference, Perimeter, Arc Length Name: 1. Cut a piece of string that perfectly fits around the outside edge of the circle. How many diameters long is the piece of string (use a marker to mark each diameter on the string)? Tape the string to this page. Tape String Here: 2. Find the Circumference of the following circles (assuming point A is the center): A. C. E. 2a. C = 2c. C = B. D. 2b. C = 2d. C = 2e. C = M. Winking Unit 4-5 page 103
3. Find the Radius of each circle given the following information: A. B. 3a. r = 3b. r = 4. Find the Arc Length of arc BC. (Assuming point A is the center) A. B. 4a. 4b. C. D. 4c. 4d. M. Winking Unit 4-5 page 104
5. Find the Arc Length of arc BC. (Assuming point A is the center) A. B. (Assume EF is tangent to the circle.) 5a. 5b. 6. Find the most appropriate value for x in each diagram. A. B. 6a. 6b. M. Winking Unit 4-5 page 105
The Babylonian Degree method of measuring angles. Around 1500 B.C. the Babylonians are credited with first dividing the circle up in to 360. They used a base 60 (sexagesimal) system to count (i.e. they had 60 symbols to represent their numbers where as we only have 10 (a centesimal system of 0 through 9)). So, the number 360 was convenient as a multiple of 60. Additionally, according to Otto Neugebauer, an expert on ancient mathematics, there is evidence to support that the division of the circle in to 360 parts may have originated from astronomical events such as the division of the days of a year. So, that the earth moved approximately a degree a day around the sun. However, this would cause problems as years passed to keep the seasons accurately aligned in the calendar as there are 365.242 actual days in a year. Some ancient Persian calendars did actually use 360 days in their year further supporting this idea. The transition to Radian measure of angle: Around 1700 in the United Kingdom, mathematician Roger Cotes saw some advantages in some situations to measuring angles using a radian system. A radian system simply put, drops a unit circle (a circle with a radius of 1) on to an angle such that the center is at the vertex and the length of the intercepted arc is the radian measure. So, a full circle of 360 is equivallent to 2π (1) radians. In the example at the right, an angle of 50 is shown. Then, a circle that has a radius of 1 cm is drawn with its center at the vertex. 1 cm 50 Arc Length 360 2 1 cm 0.873 cm Finally, the intercepted arc length is determined to be approximately 0.873 or more precisely 5π 18 radians. Similarly, it can be demonstrated the basically that 180 is equivalent to π radians. 6. Using the ratio of 180 : π convert the following degree measures to radians. a. 30 b. 80 c. 225 d. 360 7. Using the ratio of 180 : π convert the following radian measures to degrees. a. π 4 3π rads b. 10 rads c. 5π 8 rads d. 0.763 rads M. Winking Unit 4-5 page 106
8. Find the Arc Length of BC using similar circles or a fraction of the circumference. A. B. 8a. 8b. 9. Solve the following. A. Determine the perimeter B. Find an expression that would C. of the rhombus shown. represent the perimeter of the triangle. Given the perimeter of the rectangle shown below is 32 cm 2 and the length of one side is 6 cm, determine the area of the rectangle. 9a. 9b. 9c. M. Winking Unit 4-5 page 107
10. Find the perimeter of each compound figure below. A. B. Assume the compound figure includes a semicircle. (Assume all adjacent sides are perpendicular.) 10a. 10b. Assume the compound figure includes three semicircles. C. D. Assume the compound figure includes a rectangle and 2 sectors centered at point A and C respectively. 10c. 10d. M. Winking Unit 4-5 page 108
Mathematically, we can determine the value of pi using the Pythagorean Theorem.
1. Sec 4.6 Circles & Volume Circumference, Perimeter, Arc Length Name: Problem Hint Formula 1. Find the area of the rectangle and write down the formula for finding the area of a rectangle. 2. Find the area of the right triangle and write down the formula for finding the area of a right triangle. 3. Find the area of the acute triangle and write down the formula for finding the area of an acute triangle. 4. Find the area of the parallelogram and write down the formula for finding the area of an acute triangle. M. Winking Unit 4-6 page 109
Problem Hint Formula 5. Find the area of the trapezoid and write down Make a Copy the formula for finding the area of a rectangle. 6 4 Rotate Copy 8 6. Find the area of the circle below. Creates a Parallelogram exactly twice the size of the trapezoid 6 1. Find the area and perimeter of each of the following shapes. 11.5 cm 3 cm 8 cm 4 cm Area: Perimeter Area: Perimeter Area: Perimeter 1 in 7 in 6 in 9 in Area: Perimeter r: Area: Perimeter r: Area: Perimeter r:
2. Solve the following area problems. Determine the area of the rhombus shown. Find an expression that would represent the area of the triangle. Find the length of the radius given the area of the circle is 531 cm 2. Area: Area: 3. Find the following sector areas (shaded regions) using fractional parts. Sector Area : Sector Area : Sector Area : M. Winking Unit 4-6 page 111
4. Find the area of each of the shaded regions. Area: Area: 5. Solve the following problems. Find the area of the shaded region, given that AC is tangent to the circle at point B and m ABE = 70 Find the central angle of a sector that has an area of 71 cm 2 and a radius of 7 cm. Find the radius of a sector that has an area of 92 cm 2 and a central angle of 130. Sector Area : Sector Area : Sector Area : M. Winking Unit 4-6 page 112
6. Find the area of the following compound figures (assume all curved shapes are semicircles). Area: Area: 7. Find the area of the following shaded regions 4 cm Area: Area: M. Winking Unit 4-6 page 113
1. Sec 4.7 Circles & Volume Nets & Surface Areas 1. Sketch a NET of each of the following solids: Name: A. B. C. D. M. Winking Unit 4-7 page 114
2. Sketch a NET of each of the following solids: 3. Find the surface area of the following solids (figures may not be drawn to scale). Surface Area : Surface Area : M. Winking Unit 4-7 page 115
4. Find the surface area of the following solids (figures may not be drawn to scale). Surface Area : Surface Area : 12 cm 4 cm Surface Area : Surface Area : M. Winking Unit 4-7 page 116
5. Find the surface area of the following solids (figures may not be drawn to scale). Surface Area : Surface Area : Surface Area : Surface Area : M. Winking Unit 4-7 page 117
6. Solve the following problems. (figures may not be drawn to scale). Determine the Surface Area of the following rectangular prism with a missing portion. Find the Surface Area of a baseball given that its largest circumference is 23.5 cm. Surface Area : Surface Area : Determine the amount of surface area that is water on our planet in square miles. You may assume the earth is spherical, has a diameter of 7918 miles, and that water covers 71% of the Earth s surface. Given the circumference of the base of a cone is 31.4cm and the slant height is 13 cm, find the surface area of the cone. Surface Area : Surface Area : M. Winking Unit 4-7 page 118a
1. Sec 4.8 Circles & Volume Volume of Pyramids & Cones 1. Find the Volume of the following solids (figures may not be drawn to scale). Name: M. Winking Unit 4-8 page 118b
2. Find the Volume of the following solids (figures may not be drawn to scale). 12 cm 4 cm M. Winking Unit 4-8 page 119
3. Find the volume of the following solids (figures may not be drawn to scale). Using a micrometer find the volume of 6 pennies stacked directly on top of each other (which is a cylinder). Show measurements to the nearest hundredth of a millimeter. Using a micrometer find the volume of 6 pennies stacked on top of each other but so that they are slanted. Show measurements to the nearest hundredth of a millimeter. Find volume of the oblique rectangular prism. Find volume of the sphere. M. Winking Unit 4-8 page 120
4. Find the volume of the following solids (figures may not be drawn to scale). The solid below shows a gas tank for a tractor trailer truck. It is in the shape of a cylinder with a radius of 9 inches and a height of 60 inches. How many gallons of fuel will it hold if there are 231 cubic inches in one gallon? A sphere is inscribed in a cube with a volume of 27 cm 3. What is the volume of the sphere? Gallons: Find volume of the regular hexagonal prism. A snowman is created from two spherical snow balls. Given the circumference of each sphere determine the volume of the snowman. M. Winking Unit 4-8 page 121
1. Sec 4.9 Circles & Volume Volume of Pyramids & Cones Name: Using Cavalieri s Principle we can show that the volume of a pyramid is exactly ⅓ the volume of a prism with the same Base and height. Consider a square based pyramid inscribed in cube. Next, translate the peak of the pyramid. Cavalieri s Principle would suggest that the volume of the oblique pyramid is the same as the original pyramid. Next, we can create 2 more oblique pyramids with the same volume of the original with the remaining space in the cube. In this diagram, we can see the 3 oblique pyramids of equal volume pulled out from the cube. So, this demonstrates a pyramid inscribed in a cube has exactly ⅓ the volume the cube. This idea can be extended to any pyramid or cone. M. Winking Unit 4 9 page 122
1. Find the Volume of the following solids (figures may not be drawn to scale). M. Winking Unit 4 9 page 123
2. Find the Volume of the following solids (figures may not be drawn to scale). Find the volume of the regular octahedron. Find the volume of the irregular solid. The base has an area of 80cm 2 and a height of 9 cm. M. Winking Unit 4 9 page 124
3. Find the Volume of the following solids (figures may not be drawn to scale). Consider triangle ABC with vertices at A (0,0), B(4, 6), and C(0,6) plotted and a coordinate grid. Determine the volume of the solid created by rotating the triangle around the y axis. M. Winking Unit 4 9 page 125
Using Cavalieri s Principle we can show that the volume of a sphere can be found by First, consider a hemisphere with a radius of R. Create a cylinder that has a base with the same radius R and a height equal to the radius R. Then, remove a cone from the cylinder that has the same base and height. Next, consider a cross section that is parallel to the base and cuts through both solids using the same plane. Cavalieri s Principle suggests if the 2 cross sections have the same area then the 2 solids must have the same volume. The area of the cross section of the sphere is: Using the Pythagorean theorem we know: or So, with simple substitution: The area of the cross section of the second solid is: Using similar triangles we know that h = b and then, using simple substitution Volume of Hemisphere = Volume of Cylinder Volume of Cone = We also know that R = b = h. So, Volume of Hemisphere = To find the volume of a complete sphere, we can just double the hemisphere: Volume of Sphere = M. Winking Unit 4 9 page 126