Geometry/Trigonometry Summer Assignment

Transcription

1 Student Name: 2017 Geometry/Trigonometry Summer Assignment Complete the following assignment in the attached packet. This is due the first day of school. Bring in a copy of your answers including ALL WORK SHOWN, to hand in for a grade, This will count as your first test grade (10 points will be taken off for each day late). It will be graded on completeness rather than correctness. This assignment will be reviewed at the beginning of the school year and you will be tested on this material again after it is reviewed. These skills will be needed throughout the year. Make sure that you start the year off well and have your assignment ready upon your return. Please answer all questions in the packet and clearly label your answers by boxing them in. You must show work where appropriate. If you need additional space to complete problems, please do your work on a separate sheet of paper, clearly label your work, and staple your additional pages to the back of the packet.

2 Part I: 1. You work for the highway department for your county board. You are in charge of determining if and where new roads will be built. Currently there are three towns in your county. NOTE: All distances are measured in miles. a. On the coordinate graph below, place Lawrence at (3, 5), Long Beach at (3, 15), and Rockaway at (15, 1). Label each point. Connect all three towns with straight lines. b. Use the distance formula to determine which town is closest to Lawrence. Show all work. c. A new road is being built that will connect the midpoint between Long Beach and Lawrence with the midpoint between Rockaway and Lawrence. Find the midpoints, list them below and label them on the graph. Then connect the midpoints creating the new road. Show all work.

3 2. Refer to the diagram below to answer the following: a. Name an obtuse angle: b. Name an acute angle: c. Name two supplementary angles: and d. If m 3 = 63, find the measures of: 4: and 5: Points on a graph are P (4, -4), Q (1, -1) and R (-2, 2). Determine if Q is the midpoint of the line segment PR. Show all of your work.

4 Multiple Choice: Choose the correct answer and put the letter on the blank. 1. Find the next number in the sequence: { 2, 6, 18, 54, }. A. 72 B. 160 C. 162 D Find the approximate area of a circle with a radius of 12 inches. Use π = A. 452 in 2 B. 38 in 2 C. 75 in 2 D. 75 in 2 3. The area of a rectangle is 36 ft 2. Its length is 9 feet. Find the perimeter. feet A. 4 feet B. 294 feet C. 26 D. 13 feet 4. A and C are complementary. The measure of C is three times the measure of A. What is the m A? A. 90 B C. 30 D In the diagram at the right, ray BD bisects ABC. Find the value of x. A A. 12 B. -12 C. 20 D. 9 B (x + 15) (4x - 45) D C Short Answer: 6. Using either angles or segments, describe the difference between the two symbols = and.

5 Part 2: 1. Using the information given, write a two-column proof. Use the conclusions listed to fill in the blanks. A baseball diamond is shown below: GIVEN: The pitchers mound is at ft 1 90 ft m 1 + m 2 + m 3 = 180 m 1 + m 2 = 93 m 3 + m 4 = 180 PROVE: m 4 = 93 STATEMENT CONCLUSION 1. m 1 + m 2 + m 3 = m 1 + m 2 = m 3 = m 3 = m 3 + m 4 = m 4 = m 4 = Conclusion Choices: Given Substitution Property of Equality Subtraction Property of Equality

6 2. For each diagram below, find m 1, m 2 and m 3. a. b. c m 1 = m 1 = m 1 = m 2 = m 2 = m 2 = m 3 = m 3 = m 3 = 3. Find the value of x and y. 3x + 4 4y x = y = 4. Write the inverse, converse and contrapositive of the following conditional statement. If a polygon has four sides, then it is a quadrilateral. Inverse: Converse: Contrapositive:

7 5. Use the given information to find the value of x. a. Given: GM = 28 b. Given: AB BC, BC CD x + 3 2x - 8 2x + 1 4x - 5 G A M A B C D Matching: Match the statement with the property it illustrates. 1. m DEF = m DEF 2. If PO ST, then ST PQ 3. XY XY 4. If J K and K L, then J L 5. If PQ = QR and QR = RS, then PQ = RS 6. If m X = m Y, then m Y = m X A. Reflexive Property of Equality B. Symmetric Property of Equality C. Transitive Property of Equality D. Reflexive Property of Congruence E. Symmetric Property of Congruence F. Transitive Property of Congruence Short Answer 7. Can an Inverse and a Converse of the same statement be the same? Explain.

8 Part 3: 1. Use the diagram to match the angle pairs to the correct type a. 1 and 3 1. Alternate Interior Angles b. 2 and Alternate Exterior Angles c. 8 and Consecutive Interior Angles d. 6 and 7 4. Corresponding Angles e. 12 and Linear Pair f. 9 and Vertical Angles 2. On the diagram at the right, l m and a b. l m a. Which angles are congruent to 42? A b b. Which angles are supplementary to 42? Use the given diagram and the information to determine which lines, if any, MUST be parallel. a. 1 7 b. 2 and 3 are right angles. c. 2 5 d. 4 is supplementary to 6. a b c x y 2 4 6

9 4. Points A(-2,0), B(2,2) and C(4,-2) form a triangle. Determine if this is a right triangle. How do you know? Show all of your work. Multiple Choice: Choose the correct answer and put the letter on the blank. 1. Which of the following lines is parallel to y = (-5/7) x + 2? A. y + (5/7)x = -5 B. y = (7/5)x - 3 C. y = (5/7)x = 9 D. y = (-7/5)x A line m has an equation y = (-1/4) x 6. If line k is perpendicular to line m and passes through the point(5,-2), what is an equation of line k? A. y = 4x + 22 B. y = 4x - 22 C. y = (-1/4)x - 22 D. y = (-1/4)x Use the graph at right to determine which line are parallel? A. m n B. m p C. none D. n p m n p

10 Use this diagram to answer questions 4 7. l m p q 4. Which angle is supplementary to 6? A. 1 B. 10 C. 14 D. none of these 5. If m 11 = 27, then m 8 = A. 63 B. 72 C. 153 D and 14 are: A. Corresponding angles B. Supplementary angles C. Vertical angles D. Alternate Interior angles 7. If 6 9 then which lines are parallel? A. l m B. l p C. m q D. p q Short Answer: 8. Explain the difference between parallel, perpendicular and skew lines.

11 Part 4: 1. Find the measures of the numbered angles m 1 = m 2 = m 3 = m 4 = m 5 = m 6 = m 7 = m 8 = m 9 = 2. Find the measures of the numbered angles. Then classify each triangle by its angles and its sides. B A C E G D 3 4 F m 1 = m 2 = m 3 = m 4 = m 5 = m 6 = m 7 = m 8 = m 9 = Classify by Sides Classify by Angles ABC: : : : :

12 : 3. Identify the triangles in the given figure: D a. Equilateral: b. Isosceles: C c. Scalene: d. Acute; e. Right: f. Obtuse: A B Multiple Choice: Choose the correct answer and put the letter on the blank. 1. A square with a side length of 5 has one vertex at (2,0). Which of the following points cannot be a vertex of the square? A. (7,0) B. (0,7) C. (-3,0) D. (-3,-5) 2. Which postulate or theorem can be used to prove ABD CDB? A. SSS B. SAS C. AAS D. ASA D B 3. What is the value of x? A C A. 5 B. 35 3x+5 4x-10 C. 15 D What is the measure of XYZ? X A. 142 B. 128 C. 118 D. none of these Short Answer: 5. Decide whether enough information is given to prove that the triangles are congruent. Explain your answer. W 28 Y Z

13 Part 5: 1. Landscape Design: You are designing a circular pool for a triangular shaped park that is surrounded by three sidewalks. You want the center of the pool to be equidistant from the three sidewalks. a. Explain how you would find the center of the pool. b. Draw your picture below. Then use a protractor to draw the angle bisectors from each vertex of the triangle.. c. What is the name of the point of concurrency?.

14 2. Graphing: Using the graph shown below, graph the following points and connect them. M: (-1, -2) N: (11, 2) P: ( 5, 6) a. Find the midpoint of MN and MP, label them Q and R, respectively. Show your work. The coordinates of Q are: The coordinates of R are: b. Draw the two medians. Label the intersection of the two medians, T. c. Find the length of the median NR: d. Juan says that PT is 2/3 the distance of PQ. Markesha disagrees. She says that PT and TQ are equal.determine who is correct and show why.

15 3. Mapping: Wal-Mart wants to build a store that is central to the triangle below. Which point of concurrency should they use to find the center of the triangle? Multiple Choice: Choose the correct answer and put the letter on the blank. P 1. In the triangle at the right, which angle is the largest? 10 8 A. P B. Q C. S Q 5 S 2. If P is the centroid in triangle ABC and PA = 12, what is the length of EP? A A. 8 B. 18 C. 12 D. 6 F P D B E 3. If AD is the perpendicular bisector of CB and also bisects CAB, what is the value of x? A A. 31 B. 16 C. 6 D. 7 C 4x + 9 3x

16 4. Find the perimeter of triangle ABC. C D B A. 27 B. 35 C. 20 D. 23 A 5 8 B 14 C 5. If DE = 36 and G and H are midpoints of DF and EF, respectively, find the length of DH. F A. 36 B. 18 C. 9 D. 24 G D H E Short Answer: 6. Explain why the following is the converse of the perpendicular Bisector Theorem: If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. You may use the diagram below to help explain your reasoning: A o o B Explanation:

17 Part 6: 1. A concrete slump test is used to measure the water content and consistency of concrete mix. Concrete is placed in a mold for a period of time, and then the mold is removed. The distance from the top of the mold to the top of the slumped concrete is then evaluated. The shape of the cross section of the slump mold can be modeled by the points A(4,12), B(8,12), C(10,0) and D(2,0). a. What special type of quadrilateral is ABCD? How did you determine this? Show all of your work that you used to prove what type of quadrilateral the shape is. b. Describe another way to prove your answer correct. c. What is the area of the cross section?

18 2. The framework of a railroad bridge is shown below. In the diagram, GB HC, AH BJ and FBA DBC. A B C F E D G H J a. Are you given enough information to prove that BE HE? If so, explain how. b. If m FBA = 40, what are the measures of BDH and DHF? c. Suppose you were given the additional information that BH and FD are perpendicular. What could you conclude about a special quadrilateral? Explain.

19 Multiple Choice: Choose the correct answer and put the letter on the blank. 1. The diagonals of a parallelogram must A. be congruent. B. be parallel. C. be perpendicular. D. bisect each other. 2. A square is a rectangle. A. sometimes B. always C. never 3. A rhombus is a square. A. sometimes B. always C. never 4. What type of quadrilateral is shown at the right? A. trapezoid B. rectangle C. rhombus D. A, B and C 5. Find the area of a trapezoid with a height of 6 inches and bases of 4 inches and 7 inches.. A. 17 inches B. 33 square inches C. 33 inches D. 17 square inches Short Answer: 6. Explain why a square is a parellelogram, a rectangle and a rhombus. Explanation:

20 Part 7: 1. Describe what transformations must take place for shapes A, B and C in order to complete this rectangle. A B C 2. Computerized embroidery machine are used to sew letters and designs on fabric. A computerized embroidery machine can use the same symbol to create several different letters. Which of the letters are rigid transformations of other letters? Explain how a computerized embroidery machine can create these letters from one symbol. a b c d e f g h I j k l m n o p q r s t u v w x y z

21 Multiple Choice: Choose the correct answer and put the letter on the blank. 1. How many lines of symmetry does this flag of Norway have? A. 0 B. 1 C What type of rotational symmetry does this flag have? A 90 B. 180 C. none 3. An isometry is a rigid transformation. A sometimes B. always C. never 4. What type of transformation is shown at the right? A. reflection B. rotation C. translation 5. What type of transformation is shown at the right? A. reflection B. rotation C. translation Short Answer: 6. Explain why if you rotate and object then reflect it, it is a different result than if you reflected it first and then rotated it. Explanation:

22 Part 8: Find the slope between each pair of points in questions 1 to (-13, 5) (12, -6) 2. (19, -14) ( -5, -2) 3. (-7, -5) (-16, -8) 4. ( -5, -9) (35, 11) Find the slope of AB and CD and determine if the segments are parallel, perpendicular or neither, in questions 5 to A (-3, -1) B (-4, 2) C (5,5) D ( 6, 2) Slope of AB Slope of CD Segments are 6. A (8, 12) B (5, 4) C (2, 1) D ( 5, 9) Slope of AB Slope of CD Segments are 7. A (4, 1) B (6, 4) C (12, 0) D ( 10, -3) Slope of AB Slope of CD Segments are 8. A (-3, 6) B (3, 6) C (3, 2) D ( -3-4) Slope of AB Slope of CD Segments are Find the distance between each pair of points in questions 9 to (10, -3) (3, 2) 10. (-12, 3) (-3, 15) 11. (-5, -7) (21, -8) 12. ( -12, -7) ( -6, -9)

23 Find the length of AB and CD and determine if the segments are congruent or not congruent, in questions 13 and A (-3, 6) B ( 3, 2) C (3, 6) D (-3, -4) AB = CD = Congruent? 14. A (0, 0) B (3, 0) C (0, -3) D (3, -3) AB = CD = Congruent? In question 15, determine if triangle ABC is scalene, isosceles or equilateral, using the distance formula. Show work for credit. 15. A (4,4) B (6, 2) C (-2, -4) In question 16, determine if triangle ABC is a right triangle. Hint: Use slope to determine if two of the sides are perpendicular. 16. A (-7, 1) B( 4, -2) C (-3, 5) Find the midpoint between the two points in questions 17 to 20. Use the midpoint formula. Be sure to answer as an ordered pair in simplest terms. 17. (-5, -3) (25, 13) 18. (35, -2) (31, 18) 19. ( 26, -18) (-14, -14) 20. ( -16, 29) (-24, -27)

24 Find x and y in questions 21 and 22. A and B are endpoints and T is the midpoint. Use the midpoint formula. 21. A (24, y) T ( -13, 8) B ( x, -16) 22. A (x, y) T ( -3, 9) B (5, -16) 23. Find the endpoints of the midsegment of the following trapezoid. Hint: graph the points then find the endpoints using the midpoint formula. T( -5, 2) R( 2, 1) M (2, -4) A (-5, -8) Given the following vertices, graph each set of points: 24. A(2, -1) B(4, 3) C(-1, 3) D(-3, -1) What does the shape look like? Using the slope formula prove that the quadrilateral is a parallelogram. Using the slope formula prove that the diagonals are perpendicular.

25 25. A(-3, 5) B(4, 4) C(-5, 3) D(4, -4) What does the shape look like? Using slope formula prove if I am a parallelogram or a trapezoid. Am I a trapezoid or a parallelogram? 26. A( -3,-1) B(-4, 2) C(5, 5) D(6,2 ) What does the shape look like? Using the slope formula prove that the quadrilateral is a parallelogram. Using the distance formula prove that I am a rectangle by checking if my opposite sides are congruent.

26 Part 9: 1. SOLVING LINEAR EQUATIONS: Solve the equation for the indicated variable. Write your solution in the box to the right. YOU MUST SHOW ALL WORK! 1. Which value of p is the solution of? 2. What is the solution of the equation? 3. Which value of x is the solution of the equation? 4. What is the value of x in the equation? 5. Solve for x:

27 6. If and, what is the product of x and m? 7. If and, find the value of y. 8. What is the value of x in the equation? 9. If, what is the value of x? 10. Solve algebraically for x:

28 11. What is the value of x in the equation? 12. Which value of x is the solution of the equation? 13. What is the value of w in the equation? 14. In the equation, what is n is equal to? 15. Which value of x is the solution of the equation?

29 16. Solve for m: 17. Solve: 18. The number of people on the school board is represented by x. Two subcommittees with an equal number of members are formed, one with members and the other board? with members. How many people are on the school 2. OPERATIONS WITH POLYNOMIALS: Simplify each of the following according to the given instructions. 1. The sum of and can be expressed as 2. What is the sum of and? 3. The expression is equivalent to

30 4. The expression is equivalent to 5. When is subtracted from, the result is 6. What is the result when is subtracted from? 7. Find the perimeter of the triangle: 2 3x x 6x 3 8x What is the product of and? 9. What is the product of and? 10. What is the product of and? 11. What is the product of and?

31 12. The expression is equivalent to 13. The expression is equivalent to 14. What is the product of (x + 4) and ( x 4)? 15. If h represents a number, write an equation that is a correct translation of "Sixty more than 9 times a number is 375". 16. The width of a rectangle is 4 less than half the length. a. If represents the length, write an equation that could be used to find the width, w. If the area of the rectangle is 40, write an equation that could be used to find the area of the rectangle in terms of l.

32 1. 3. FACTORING COMPLETELY: Factor each of the following expressions completely. Remember to look for your factoring methods in the following order: i. Greatest Common Factor ii. Trinomial A/M iii. Factoring by Grouping ( Tricky Trinomial) iv. Difference of Two Perfect Squares x x + 7x + 10x 2. 3 x x x 6x x 10x y a 2b x + x a ba by 13. z 4z 21z y 4y

33 4. SOLVING QUADRATIC EQUATIONS BY FACTORING: Solve each of the following quadratic equations. YOU MUST SHOW ALL WORK. Remember: i. Set the equation equal to 0. ii. Factor the non-zero side of the equation completely iii. Set each factor equal to 0. iv. Solve each equation. 1. What is the solution set of the equation 3x 2 = 48? 2. What is the solution set of the equation x 2 5x = 0? 3. 2 x 4x = x(x 1) = 5 5. x = 2 x 6. 2 x = x

34 7. x+ 3 x = 8 x 3 8. y + y = y b 6b 1 + = y + y = 11. The length of a rectangular window is 5 feet more than its width, w. The area of the window is 36 square feet. Find the dimensions of the window. 12. The length of a rectangle is 2 feet less than twice the width. The area of the rectangle is 84 square feet. Find the dimensions of the rectangle.

35 5. GRAPHING ON THE COORDINATE PLANE: 1. What is an equation of the line that passes through the points and? 2. What is the slope of the line represented by the equation? 3. What is the solution to the system of equations? (Where do the two graphs intersect?) 4. The lines represented by the equations and are i. Parallel, but not the same line ii. Perpendicular iii. Intersecting but not perpendicular iv. The same line 5. Write an equation of a line that is parallel to the line whose equation is and that passes through the point.

36 6. Write an equation of a line that is perpendicular to the line whose equation is and that passes through the point. 7. The slope of is and the slope of is. If, determine and state the value of x. 8. Write an equation that represents the line that passes through the points and. 9. What is the slope of a line passing through points and? 10. Write an equation of a line that represents a VERTICAL line. 11. Write an equation of a line that represents a HORIZONTAL line. 12. Graph the following two functions and determine the points where they intersect.

37 13. Point lies on the line whose equation is. What is the value of k? 14. Sketch the graph of y = 4 and x = -3 on the same set of axes. Make sure to label each equation.

38 6. SQUARE ROOTS: Please show all work in the space provided. Box in your answer. 1. What is expressed in simplest radical form? 2. Express in simplest radical form

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