Math 3 Plane Geometry Part 3 Unit Updated July 28, 2016

Transcription

1 Reviewing area and circumference of circles Area of a circle = (memorize this formula if you haven't already done so) Circumference of a circle = (memorize this formula if you haven't already done so) 1. The figure below shows 4 congruent circles, each tangent to 2 other circles and to 2 sides of the square. If the length of a side of the square is 40 cm, then what is the area, in square cm, of 1 circle? 2. Circles with centers G and K intersect at points C and F, as shown below. Points B, G, H, J, K, and D are collinear. The lengths of AC, CE, and HJ are 30, 20, and 5 cm respectively. What is the length, in centimeters, of BD? 3. A costume designer wants to put a while silk band around a cylindrical top hat. If the radius of the cylindrical part of the top hat is 5 inches, how long, in inches, should the white silk band be to just fit around the top hat? (leave answer in terms of ) 1

2 4. A museum recently acquired a new statue for its collection. Museum security wants to mark a shape on the floor to keep visitors of the museum a minimum of 6 feet away from the statue in all directions. Which of the following shapes would allow visitors to stand 6 feet away from the statue on all directions? A. A circle with a radius of 6 feet B. A square with 6 foot sides C. A triangle with 6 foot sides D. A circle with a diameter of 6 feet E. A pair of parallel lines, each 6 feet from the statue Circle arc length and sector area "Arc length" is a fraction of the circle's circumference: circumference "Sector area" is a fraction of the circle's area area Arc length example In the figure shown, the radius is 5 and the measure of the central angle is 72 The arc length is, or of the circumference: = = Sector area example In the figure below, the radius is 6 inches and the measure of the sector's central angle is 30 The sector has, or, of the area of the circle: or approximately 9.42 square inches. 2

3 5. What is arc length AB in the circle at the right? 6. The youth center has installed a swimming pool on level ground. The pool is a right circular cylinder with a diameter of 30 feet and a height of 6 feet. A diagram of the pool and its entry ladder is shown below and to the left. A plastic cover is made for the pool. The cover will rest on the top of the pool and will include a wedge-shaped flap that forms a 45 angle at the center of the cover, as shown in the figure below and to the right. A zipper will go along one side of the wedge-shaped flap and around the arc. What is the length of the zipper, in feet rounded to the nearest tenth of a foot? 7. Sam ordered a 14" (diameter) pizza. He ate 2 slices leaving an empty space with a 120 central angle. How much pizza, measured in square inches, did Sam eat? 3

4 Volume and surface area Volume of a cylinder =, which is still just the area of the base (a circle) times the height. 8. What is the volume of the cylinder to the right? Surface area of a cylinder = 2πrh + 2π It is the top and bottom (area of a circle times 2) + the area of the side, which is a rolled up rectangle. The length of the rectangle is the same as the circumference of the circle. Think of unrolling the label off a soup can. 9. What is the surface area of the cylinder in question 8? 10. Joseph has a container to hold quarters. The container is shaped like a cylinder and each quarter has a thickness of 1.75 mm and a diameter of mm. If the container is 70 mm tall, what is the maximum number of quarters that can fit in the container? (Hint: is the volume of a cylinder really necessary information to answer this question?) 4

5 Volume of a cone =. It's the same as the volume of a right cylinder divided by 3. When asked to solve something with a cone, the formula will also be given so no need to memorize this one, just know how to use it. 11. The volume of a cone, which is derived by treating it as a pyramid with infinitely many lateral faces, is given by the formula V =, where r is the radius of the base, and h is the height. If the radius is 4 and the height is 8 what is the volume of the cone? Leave answers in terms of Volume of a sphere = Surface area of a sphere = Hint: When asked to solve volume or surface area of a sphere, the formula will also be given so no need to memorize these, just know how to use them. 12. If the volume of a sphere is 288 cubic inches, then what is the surface area, in square inches of the same sphere? (Note: for a sphere with radius r, the volume is and the surface area is ) [Hint: to find the cube root on a calculator, look for a button that looks like this On most calculators it requires using the "2nd" button. 13. The radius of a sphere is meters. What is the volume of the sphere, to the nearest cubic meter? Use the formula V = 5

6 Diagonal of a cube - you can find the diagonal of a cube in one of two ways: remembering the formula or working it out by using using the Pythagorean theorem twice. In the diagram below, line b is a diagonal of one of the sides. This diagonal makes the base leg of a new triangle with hypotenuse d, which is the diagonal of the cube. The formula for finding the diagonal of any rectangular solid is d = where In this case, since it's a cube and the length, breadth, and height are all the same = so or = and if you take the square root of both sides 14. What is the diagonal of a 4 inch cube? (leave answer in terms of square roots) 15. A sphere is inscribed in a cube with a diagonal of ft. In feet, what is the diameter of the sphere? Special triangles When using the Pythagorean theorem we often get answers with square roots or long decimals. There are a few special right triangles that give integer answers. We've already talked about the right triangle, now it's time to learn a few others that also give integer answers. When you are able to recognize these kinds of triangles, you don't even have to use the Pythagorean theorem, which makes things simpler and easier. If you forget or don't recognize these, no worries, just use the Pythagorean theorem, it works every time. The special right triangles are the 3-4-5, the , the , and the On the figures above the "k" by each number just means that multiples of these ratios work as well. 6

7 16. The figure to the right is a right triangle. What is the length of the base leg? 17. The figure at right is a right triangle. What is the length of the hypotenuse? 18. The figure below is a right triangle. Solve for a. More on polygons. The term "polygon" is a combination of two Greek words, "poly" which means "many" and "gonia" which means "angle." A polygon is a two-dimensional closed figure bounded by straight line segments. Most of the basic shapes, such a triangles, squares, rectangles, etc. are examples of polygons. Circles are not polygons. They don't have straight sides or angles. 7

8 Polygons are named by how many sides or angles they have. Triangles have 3 sides, quadrilaterals have 4 sides, pentagons have 5 sides, etc. The shapes that we usually think of are called "regular" polygons, but polygons can be shaped irregularly as well. Interior angles of a polygon We know that the sum of the interior angles of a triangle equal 180. We can use this information to find the sum of the interior angles of other polygons. A square (or any quadrilateral) can be divided into 2 triangles. Since the interior angles of each of the triangles will be 180, the total will be 360. A pentagon can be divided into 3 triangles. Since the interior angles of each of the triangles will be 180, the total for the pentagon will be

9 Likewise, a hexagon can be divided into 4 triangles with a total of 720. Using this same pattern we can figure out the sum of the measure of the interior angles of any polygon. If n is the number of sides of the polygon, then the sum of the measures of the interior angles is 19. What is the sum of the interior angles of this irregular octagon? 20. What is the measure of angle x in the figure below? 9

10 21. The following figures show regular polygons and the sum of the degrees of the angles in each polygon. Based on these figures, what is the number of degrees in an n-sided regular polygon? A. 180 B. 180 C. 60 D. 20 E. Cannot be determined from the information given. Interior angles of "regular" polygons Regular polygons are shapes where all of the angles are the same and all of the side lengths are the same. We can determine the sum of the interior angles of any polygon. If the angles are all the same, we can simply divide the total by the number of angles to find the measure of one of the angles. The formula is. 22. What is the measure of each angle in this regular octagon? 23. What is the measure of angle C in the regular polygon to the right? 10

11 24. The figure below is a regular polygon, what is the measure of angle X? 25. In the figure below, 2 nonadjacent sides of a regular pentagon (5 congruent sides and 5 congruent interior angles) are extended until they meet at point X. What is the measure of angle X? Similar polygons We've already learned about similar triangles and how they are proportional. Similar polygons are also proportional, so you can set up a ratio to solve for missing information. All squares are similar, can you think why that would be? 26. Given parallelogram ABCD below and parallelogram EFGH (not shown) are similar, which of the following statements must be true about the two shapes? A. Their areas are equal. B. Their perimeters are equal. C. Side AB is congruent to side EF. D. Diagonal AC is congruent to diagonal EG. E. Their corresponding angles are congruent. 11

12 27. Rectangles ABCD and EFGH shown are similar. Using the given information, what is the length of side FG, to the nearest tenth of an inch? 28. Two similar triangles have perimeters in the ratio 3: 4. The sides of the smaller triangle measure 3 in, 4 in, and 5 in. What is the perimeter, in inches, of the larger triangle? Geometric sequence - means a pattern of multiplication, in which each term is multiplied by the same amount to determine the next term. In a geometric sequence, each term divided by the prior term yields a constant, or common, ratio. For example 3, 6, 9, 12, is a geometric sequence since each term is multiplied by 3 to get the next term. The Fibonacci sequence is an example of a geometric sequence where each new number is approximately 1.6 times the previous number. 29. The degree measures of the 4 angles of quadrilateral LMNO, shown at right, form a geometric sequence with a common ratio of 2. What is the measure of angle N? (Hint: remember that a quadrilateral has a sum of 360 for all 4 angles. Call the smallest angle x, then the next smallest angle will be 2 times x, and the next will be 4x and the biggest will be 8x.) 12

13 Arithmetic sequence An arithmetic sequence goes from one term to the next by adding the same number each time. For example 1, 3, 5, 7, 9, 11,... is an arithmetic sequence since we add 2 each time to get the next term in the sequence. The angles in a triangle follow a arithmetic sequence. The smallest angle is 30, then add 30 for the next angle and add 30 again to get the measure for the last angle. Questions reviewing perimeter and area 30. What is the area, in square inches, of a square with a side length of 6 inches? 31. The rectangular soccer field at the Recreational Park is twice as long as it is wide. The perimeter of the field is 300 yards. What is the width, in yards, of the soccer field? 32. In the following figure, all interior angles are 90, and all dimension lengths are given in inches. What is the perimeter of this figure in inches? 13

14 33. All line segments that intersect in the polygon below do so at right angles. If the dimensions given are in inches, that what is the area of the polygon, in square inches? (Hint: Find the area of a 10 x 12 rectangle as if the cutouts weren't there and then subtract the area of the smaller rectangles.) 34. The six-sided figure below is divided into 8 congruent isosceles right triangles. The total area of the 8 triangles is 36 square inches. What is the perimeter, in inches, of the figure? (leave answer in terms of square root as applicable) 35. In the figure below, the top and bottom of the rectangle are tangent to the circle as shown. The rectangle has a length of 4 and width of 2. What is the area of the shaded region? A B C D. 8 - E. 8-14

15 36. For a decorating project, Aubrey found the area and perimeter of a drawing she made of a beach scene. She found that the area of her rectangular drawing was 144 square inches and that the perimeter was 80 inches. When she arrived at the craft store to purchase a frame for her drawing, she discovered that she had forgotten to write down the dimensions of her drawing. What are the dimensions of Aubrey's drawing, in inches? (hint: Whenever multiple choice answers are given you can use them to work backwards. Plug in the possible answers and see which one works, but remember it has to work for both the area and the perimeter) A. 4 by 36 B. 6 by 24 C. 8 by 18 D. 9 by 16 E. 12 by The graph of is shown in the standard (x,y) coordinate plane below for values of x such that The x coordinates of points D and E are both 4. What is the area of DEO, in square coordinate units? (Hint: The trick in this question is ignoring all irrelevant information. It's really asking for the area of DEO, ignore everything that doesn't matter.) 38. In the following figure, AB is perpendicular to BC. The lengths of AB and BC, in inches, are given in terms of x. Which of the following represents the area of triangle ABC, in square inches, for all x 1? A. B. C. D. E. 15

16 Questions reviewing angles, Pythagorean theorem, and similar triangles 39. The isosceles triangle in the quilt square below has one angle that measures 40 What is the measure of each of the other two angles in the triangle? 40. In the figure below, line m is parallel to line n, point A lies on line m, and points B and C lie on line n. If angle BAC is a right angle, what is the value of y? (Hint: you can't assume that angles that look the same have the same angle, use what you know about parallel lines and transversals.) 41. Right triangle QRS is isosceles and has its right angle at point R. Point T is collinear with points R and S, with S between R and T. What is the measure of angle QST? (Hint: Draw a figure. Collinear means "in line with" so point T in on the same line as points R and S) 16

17 42. In the figure below, AD is perpendicular to BD, AC is perpendicular to BC, and AD BC. Which of the following congruencies is NOT necessarily true? A. AC BD B. AD AE C. AE BE D. Angle DAB Angle CBA E. angle EAB Angle EBA 43. How many units long is one of the sides of a square that has a diagonal 16 units in length? (leave answer in terms of square roots) 44. In the figure below, A, D, B, and G are collinear. If angle CAD measures 70, angle BCD measures 45 and angle CBG measures 130 what is the degree measure of angle ACD? 45. In the following figure, AD, BE, and CF all intersect at point G. If the measure of angle AGB is 60 and the measure of angle CGE is 110, what is the measure of angle AGF? 17

18 46. In the figure below, parallel lines r and s are cut by transversal t. The measure of angle p is 20 less than three times the measure of angle q. What is the value of p - q? (Hint: underline or circle the actual question so you don't mistakenly think you're done when you find q.) 47. In the following figure, BD bisects angle ABC. The measure of angle ABC is 100, and the measure of BCD is 25. What is the measure of angle BDA? 48. The triangle XYZ, that is shown below has side lengths of x, y, and z inches and is not a right triangle. Let X' be the image of X when the triangle is reflected across YZ. Which of the following is an expression for the perimeter, in inches, of quadrilateral X'YXZ? (Hint: draw a diagram of the newly created quadrilateral.) A. 2(y + x) + x B. 2(x + y + z) C. 2(x + y) D. 2(x + z) E. 2(y + z) 18

19 49. In the figure below, PQ intersects triangle ABC at points E and D. Angle ABC measures 80, angle SAB measures 130, and angle CDQ measures 110. What is the measure of angle BEP? 50. In DEF, x represents the measure of angle EDF. The measure of angle DEF is 30 greater than the measure of angle EFD, and the measure of angle EFD is 15 less than the sum of the measures of angle EDF and DEF. Which of the following expressions represents the measure of angle EFD? (Hint: draw a triangle and label it, then put the value of each angle as described.) A. B. C. D. E. 51. In the figure below, the measure of angle QPT is 100. If the measure of angle RPT is (Y + 17), what is the measure of angle QPR? A. (117 - Y) B. (117 + Y) C. (Y - 83) D. (83 - Y) E. (83 + Y) 19

20 52. In the figure below, PS is parallel to QR, and PR intersects SQ at T. If the measure of angle PST is 60 and the measure of angle QRT is 30, then what is the measure of angle PTQ? 53. In the following figure, line t crosses parallel lines m and n. Which of the following statements must be true? A. d = c B. a = d C. b = e D. f = g E. d = h 54. In the figure below, JK is parallel to MN, and JM and KN intersect at L. Which of the following statements must be true? A. Triangle JKL is congruent to triangle MNL B. JL is congruent to LM C. JK is congruent to MN D. JM bisects KN E. Triangle JKL is similar to MNL 20

21 55. A circle is inscribed in a square, as shown below. If x is the distance from the center of the circle to a vertex of the square, then what is the length of the radius of the circle, in terms of x? (Hint: The choice "cannot be determined from the information given" is almost never the right answer on the ACT math section) A. B. C. D. E. Cannot be determined from the information given 56. Two girls walk home from school. Starting from school, Susan walks north 2 blocks and then west 8 blocks, while Cindy walks east 3 blocks and then south 1 block. Approximately how many blocks apart are the girls' homes? (Hint: draw a diagram and use the Pythagorean theorem) 57. In the rhombus below, diagonal AC = 6 and diagonal BD = 8. What is the length of each of the four sides? (Hint: Remember that the diagonals of a rhombus are perpendicular bisectors of each other) 21

22 Answers: c, A feet 7. or or or or A E in inches yards inches E 36. A E B E E 51. D E 54. E 55. D blocks