Geometry Summer Assignment

Transcription

1 Geometry Summer Assignment You must show all work to earn full credit. This assignment will be due Friday, August 24, It will be worth 50 points. All of these skills are necessary to be successful in Geometry Honors. Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. However, in order to solve those questions, algebra will be crucial. There will be a test on these topics at the beginning of the year. Please follow criteria for credit shown below. You may write on the back of your notebook paper. Do NOT write on this worksheet. -Mrs. Fussner

2 Section 1: Expressions & Equations A. Integer Operations - Integers are positive & negative whole numbers. No Calculators Same Signs + and + and Addition Subtraction Multiplication Division Add the numbers Take the sign of both numbers Examples = 14 ( 4) + ( 5) = 9 Different Subtract the numbers signs Take the sign of the + and number with the larger and + absolute value Examples 14 + ( 24) = 10 ( 9) + 10 = 1 Subtract the numbers Take the sign of the number with the larger absolute value = 5 ( 7) ( 15) = ( 7) + 15 = 8 Add the numbers Take the sign of the number with the larger absolute value 10 ( 5) = = 15 ( 8) 7 = 15 Multiply the numbers The product is always positive. (7)(8) = 56 ( 11)( 12) = 132 Multiply the numbers The product is always negative (8)( 12) = 96 ( 12)(10) = 120 Divide the numbers The quotient is always positive. 9/3 = 3 ( 72)/( 8) = 9 Divide the numbers The quotient is always negative. 9/( 3) = 3 ( 72)/( 8) = ( 3) ( 8) ( 7) ( 1) 8. 3( 4) ( 18) ( 4) ( 27) B. Order of Operations No Calculators Example: ( 12 3) (7 + 4) (3 2 6) ( 6 3) (12 15) ( 6) + 8 ( 2) Evaluate each expression if a = 12, b = 9, and c = a + 2b c c3 ab (a b) 2 5c C. Solving Equations - Solve each equation for x. You may use a calculator but MUST show every step = 4x 6x 27. 8x 2 = 9 + 7x 29. 4x + (5x 36) = = 4( 6x 3) 28. 3(4x + 3) + 4(6x + 1) = (3x 5) + (2x 10) = 180

3 D. Solving Equations by Clearing the Fraction You may use a calculator but MUST show every step x x = 3 + x ( 2x + 1) = ( 3x + 1) = E. Solving Systems of Equations Systems may have zero solutions, one solution or infinitely many solutions. Elimination Example: Solve the following systems of equations by substitution. You may use a calculator but MUST show every step. 34. x + 12y = x + 2y = 6 x = 8y 12 x 2y = 10 Solve the following systems of equations by elimination. You may use a calculator but MUST show every step x + 5y = x + 6y = 0 3x y = 11-7x + 2y = 31

4 Section 2: The Coordinate Plane & Linear Functions F. The Coordinate Plane Tell what point is located at each ordered pair. 38. (3, 2) 39. ( 7, 8) 40. (2, 3) 41. ( 4, 4) 42. ( 5, 5) 43. ( 5, 0) Write the ordered pair for each given point & name the quadrant it is in. 44. E 45. M 46. C G. Slope Find the slope in each problem ( 7, 8) 52. (6, 9) (3, 9) 53. 2x + 3y = x = 4

5 H. Graphing & Writing Equations of Lines Slope-intercept form y = mx + b Point-slope form y y 1 = m(x x 1) For each set of ordered pairs, calculate the slope and write the equation of the line passing through each of the points in both slope-intercept & point-slope form. Then graph the equation. 55. (0, 3) and (5, 1) 56. (-1, 3) and (-4, 7) 57. (9, -2) and (9, 4)

6 I. Parallel & Perpendicular Lines Write an equation that is parallel to the given equation. 58. y = 3x x + y = 3 Write an equation that is perpendicular to the given equation y = x y 4x = Choose the two parallel lines 63. Choose the two perpendicular lines y - 6 = 3x y 3x = -4 y + 10 = -4x y = 4x y = -x y + 9 = -3x -4y = x 4y = x Write an equation in slope-intercept form for the line that passes through the point (8, -12) and is parallel to 3 y = x Write an equation in slope-intercept form for the line that passes through the point (4, -3) and is perpendicular to y = 4x +5 J. Midpoint & Distance Formula Use the midpoint and distance formulas to find the midpoint and distance between each pair of points: 66. (7, 11) and (-1, 5) 69. (2, 0) and (8, 6) 67. (-2, -1) and (3, 11) 70. (-2, -6) and (6, 9) 68. (-10, 2) and (-7, 6) 71. (-3, 2) and (6, 5)

7 Section 3: Radicals Radicals or roots are the opposite operation of applying exponents. You will undo exponents by using a radical. Index 3 x 4 Radical Symbol Radicand Read as the cube root of x 4 K. Perfect Squares 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400,. *This means if you see any of these numbers under the radical you can quickly simplify it by finding the number that multiplies by itself to get the number. Simplify: L. Non-Perfect Squares When the number under the radical cannot be equally split in this case we have to reduce it to its lowest terms. Math Look Fors Identify the largest perfect square that divides evenly into the radicand Take the square root of the perfect square radical and leave the non perfect square under its radical Simplify:

8 M. Radicals with Variables Math m 5 m 4 m 1 m 2 m Look Fors Break the variable down into it s perfect square exponent (even exponent) and remaining exponent (remember that multiplying 2 bases that are the same means we add the exponents) Take the square root of the perfect square variable by dividing its exponent by 2 and keep the non perfect square variable under its radical. Simplify: x t p x 6 y 5 z 7 N. Rationalizing the Denominator A radical cannot be in the denominator of a fraction, so in order to fix it we have to multiply both the numerator 2 and denominator by the radical in the denominator (or one that will create a perfect square). Example: 3 Math Look Fors = Multiply both the numerator and denominator by the radical in the denominator Simplify the radicals to solve Simplify: Section 4: Proportions O. Solving Proportions Solve for x: 92. x 7 = x = x 7 = x+1 10 = = x 2 x = 15 x

9 P. Conversions Convert to the given unit: ft. = in m = cm cm = m in = ft ft 2 = in in 2 = ft m 2 = cm cm 2 = m 2 Section 5: Polynomials Q. Multiply Polynomials *Distribute or use FOIL method. Product of First Terms Product of Outer Terms Product of Inner Terms Product of Last Terms Find each product. (x + 5)(x + 7) = (x)(x) + (x)(7) + (5)(x) + (5)(7) = x 2 + 7x + 5x + 35 = x x (r + 1)(r 2) 108. (n 5)(n + 1) 107. (3c + 1)(c 2) 109. (2x 6)(x + 3) R. Factoring Factoring is the process of un-doing a polynomial. Factors are what items multiplied together to get a product. First always check to see if you can factor out a GCF. Math t 2 + 8t + 12 Look Fors t 2 + 8t + 12 Identify the factors of c 1 12, 2 6, 3 4 are factors of c 6 and 2 can be added to get 8 or b Find the factors of c that add or subtract to equal b (t )(t ) Create your factors (t + )(t + ) Identify the signs that fit into factors If + and +, then factors are both + If + and, then larger factor get + and smaller factor gets If and, then larger factor gets and smaller factors get + If and +, then both factors are - (t + 2)(t + 6) Plug in the numbers Factor each polynomial if possible. If the polynomial cannot be factored using integers, write prime p 2 + 9p g 2 7g n 2 + 3n y 2 5y t 2 + 9t r r 48

10 Section 6: Solving Quadratic Equations S. Solving Quadratic Equations by Factoring Example: x 2 6x + 8 = 0 Math x 2 6x + 8 = 0 Look Fors Identify the factors of c 1 8, 2 4 are factors of c 2 and 4 can be added to get 6 or b Find the factors of c that add or subtract to equal b (x )(x ) Create your factors (x )(x ) Identify the signs that fit into factors (x 2)(x 4) = 0 Plug in the numbers x 2 = 0 x 4 = 0 Set each factor equal to zero x 2 = 0 x 4 = 0 Solve for x in each problem using inverse operations x = 2 x = 4 x can either have a value of 2 or 4 Identify the solution Solve each equation by factoring. Check the solutions d 2 + 7d + 10 = y 2 2y 24 = 0 T. Solving Quadratic Equations by Completing the Square Solve by completing the square: 118. x 2 + 4x - 10 = x x - 4 = 0 U. Solving Quadratic Equations by Quadratic Formula 120. x 2 + 6x + 1 = x 2-12x + 30 = 0 Solve using quadratic formula: 122. x 2-11x + 7 = x 2 + 7x - 4 = x 2-12x - 9 = x 2 + 6x + 20 = 0

11 Section 7: Geometry Basics V. Points, Lines & Planes P or point P AB or BA XY or YX PQ Plane EFG or Plane T 126. Name any two line segments Which point is not coplanar (points all on the same plane) with the points U and V? 128. Write a set of points with are collinear (points on the same line) Name a pair of opposite rays (have the same endpoint & extend in opposite directions) Name the plane in two ways. W. Angles Name each angle 4 ways, classify it and give its exact measure using a protractor

12 134. Name two acute vertical angles Name two obtuse vertical angles Name a pair of adjacent angles 137. Name a linear pair Name a pair of complementary angles Name an angle supplementary to FGE

13 X. The Pythagorean Theorem Find the missing side length of the right triangle using The Pythagorean Theorem c 8 x a 26 6 x Y. Transformations Name the type of transformation depicted in the diagram below. If a reflection, state the line of symmetry. Dashed figure (preimage) solid figure (image)

14 Z. Perimeter, Area, Surface Area & Volume Lateral Area (LA) area of the sides Surface Area area of sides & bases Key: Prism Pyramid Cylinder Cone Sphere Ph 1 2 Pl 2πrh πrl LA + 2B LA + B LA + 2B LA + B 4πr 2 Volume Bh 1 3 Bh Bh 1 3 Bh 4 3 πr 3 P = perimeter of the base of the solid h = height of solid (or altitude) l = slant height of pyramid or cone (might have to use pythagorean theorem to find this) r = radius of circle B = area of the base of the solid Determine which choice BEST describes the figure Find the surface area and volume of the figure