Name. Geometry Honors. Unit 1 Coordinate Geometry. Practice Packet

Transcription

1 Name Geometry Honors Unit 1 Coordinate Geometry Practice Packet 1

2 Slope, // Lines and Lines 1. Determine whether the lines with equations y x 1 and 1 y x 5 are parallel, perpendicular or neither. Explain your reasoning.. What is the slope of a line whose equation is y x 3 0? What is the y-intercept of this line? 5 3. Given lines with the equations y ( x 5) and y x 6, determine whether these lines are parallel, 5 perpendicular or neither. Explain your reasoning. 4. What is the slope of AB when the coordinates of A are (3, 6) and the coordinates of B are (5, -)? What is the slope of a line perpendicular to AB? 5. Write an equation of a line passing through the point (5, -) and parallel to the line whose equation is y x 4. Graph your line, and the line whose equation you wrote.

3 6. Write an equation of the line that contains the points (-1, -) and (3, ). 7. Write an equation of the line that contains the points (3, 8) and (9, 6). 8. Write an equation of the line that passes through the point (, 3) and is parallel to the line whose equation is x y. 9. Write an equation of the line that passes through the point (, 3) and is parallel to the line whose equation is x y Write an equation of the line that passes through the point (6, 7) and is perpendicular to the line whose equation is y 3x 1. 3

4 11. Write an equation of the line that passes through the point (8, 1) and is perpendicular to the line whose equation is x y Write an equation of the line that passes through the point (6, ) and is perpendicular to the line whose equation is y Write an equation of the line that passes through the point (, 3) and is parallel to the line y. 14. Write an equation of the line that passes through the point (, 3) and is perpendicular to the line x. 15. Write an equation of the line that passes through the point (, 3) and is perpendicular to the line whose equation is x y. 4

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9 Distance Directions (#1-10): Determine the distance between each pair of points. When necessary, express your answer in simplest radical form. If you would find it helpful, you may use a sheet of graph paper. 1. (-5, -3) and (4, 6). (-7, 5) and (3, 0) 3. (-, 1) and (-5, 5) 4. (8, -3) and (9, -10) 5. (4, 6) and (8, 6) 6. (-3, -) and (-3, 3) 7. (4, 3) and (7, 6) 8. (-, ) and (-4, 5) 9

10 9. (c, d) and (0, 0) 10. (a + b, b) and (a, 0) Express your answer in terms of c and d Express your answer in terms of a and b 11. The point (5, 4) lies on a circle. If the center of the circle is at (3, ), what is the length of the radius? 1. Two birds are flying toward a birdhouse that is midway between them. The birds are located at A(-4, 4) and B (10, -). If each grid on the graph represents 100 feet, how far will each bird fly to arrive at the birdhouse? Round your answer to the nearest tenth. 13. Use the distance formula to find the perimeter of a triangle whose vertices are A(-, ), B(10, ) and C(10, 7). 10

11 14. The points X (0, 0), Y (3, 4) and Z (-1, 1) form a triangle. Is the triangle isosceles? Justify your answer. Directions (#15-16): Find all possible values of t given the following conditions. 15. The distance from (t, 0) to (0, 5) is The distance from (t, t) to (5, 0) is 5. 11

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13 Midpoint 1. A line segment has endpoints (1, -) and (-4, 5). What are the coordinates of the midpoint of this segment?. The midpoint of a line segment is (-1, -1). If the coordinates of one endpoint are (-3, 1), what are the coordinates of the other endpoint? 3. A line segment has endpoints (-5, 6) and (-1, -7). What are the coordinates of the midpoint of this segment? 4. The midpoint of a line segment is (-3.5, -1). If the coordinates of one endpoint are (5, -), what are the coordinates of the other endpoint? 5. A line segment has endpoints (0.35, 3.8) and (-4.5, -5.4). What are the coordinates of the midpoint of this segment? 13

14 6. The midpoint of a line segment is (-5.8,.4). If the coordinates of one endpoint are (-5, 6), what are the coordinates of the other endpoint? A line segment has endpoints (, ) and (, ). What are the coordinates of the midpoint of this segment? The midpoint of a line segment is (, ). If the coordinates of one endpoint are (-3, 1), what are the 4 coordinates of the other endpoint? 9. A line segment has endpoints ( x 1, y) and ( 3x 5, 3y). What are the coordinates of the midpoint of this segment? 10. A line segment has endpoints ( a 6, 3b) and ( a 4, 9b 4). What are the coordinates of the midpoint of this segment? 11. Is it possible for an angle to have a midpoint? How about rays? Lines? Explain or justify your response. 14

15 1. You are asked to find the length of a line segment. Circle the formula that would be most appropriate to use. DISTANCE MIDPOINT Explain your choice: 13. You are asked to find the place that marks the coordinates of the middle of a line segment. Circle the formula would be the most appropriate to use. DISTANCE MIDPOINT Explain your choice: 14. You have been asked to find the perimeter of a triangle with vertices (-, 1), (4, 5) and (0, 6). Circle the formula that would be the most appropriate to use. DISTANCE MIDPOINT Explain your choice: 15. You have been asked to determine whether or not a line segment with endpoints (0, 5) and (1, 8) is congruent to a line segment with endpoints (, 6) and (3, 11). Circle the formula that would be the most appropriate to use. DISTANCE MIDPOINT Explain your choice: 16. You have been asked to determine the coordinates of the point that divides a line segment into two congruent parts. Circle the formula that would be the most appropriate to use. DISTANCE MIDPOINT Explain your choice: 17. If you know the coordinates of the endpoints of the diameter of the circle, which formula could be used to find the coordinates of the center of the circle? Circle the formula that would be the most appropriate to use. DISTANCE MIDPOINT Explain your choice: 18. If you know the coordinates of the endpoints of the diameter of the circle, which formula could be used to find the length of the diameter? Circle the formula that would be the most appropriate to use. DISTANCE MIDPOINT Explain your choice: 15

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17 Perpendicular Bisectors 1. XY bisects LN at point M. If LM x 45 and MN 5x 36, solve for x and find LN.. Write an equation of the perpendicular bisector of the line segment with endpoints (8, ) and (-6, 4). 3. Write an equation of the perpendicular bisector of the line segment with endpoints (7, 3) and (3, 7). 4. Write an equation of the perpendicular bisector of the line segment with endpoints (, 4) and (-10, 4). 17

18 5. Write an equation of the perpendicular bisector of the line segment with endpoints (-, 1) and (8, 3). 6. AB bisects CD at point E. If AE 5x 56, BE 3x 194, CE 4y 33 and DE 3y 184, find the length of CD. 7. Write an equation of the perpendicular bisector of the line segment with endpoints (-1, ) and (3, 5). 8. Write an equation of the perpendicular bisector of the line segment with endpoints (0, 3) and (5, 8). 18

19 9. Write an equation of the perpendicular bisector of the line segment with endpoints (-, 3) and (7, -4). 10. Write an equation of the perpendicular bisector of the line segment with endpoints (6, -3) and (6, 5). 11. Write an equation of the perpendicular bisector of the line segment with endpoints (-10, -4) and (-, 8). 19

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21 Partitioning a Segment 1.. Given the points A (, 5) and B (5, 8), determine the coordinates of point P on directed line segment AB that partitions AB into the ratio 1:. 1

22 3. Point P partitions directed line segment AB with A(-1, ) and B (7, 8) into a ratio of 1:3. Find the coordinates of point P. 4. Point P partitions directed line segment AB with A(-4, ) and B (, 8) into a ratio of 1:. Find the coordinates of point P.

23 5. Point P partitions directed line segment AB with A(1, -5) and B (9, -1) into a ratio of :3. Find the coordinates of point P. 6. Point P partitions directed line segment AB with A(-, 4) and B (5, 10) into a ratio of 3:. Find the coordinates of point P. 3

24 7. C is a point located on AB such that AC:CB = 4:1. If A(0, 4) and C(8, -), find the coordinates of B. 8. Given A(-6, -4) and B(0, 8) on directed line segment AB, find point P on the segment such that AP=3PB. 4

25 9. A map shows a straight jungle path between two villages. As the rainy season approaches, the villages decide to establish two shelters such that the shelters divide the path into three equal parts. Find the coordinates of the points at which the rest stops should be built if the villages are located at (-3, -6) and (3, 9). 10. Find the length and the midpoint of the segment whose endpoints are (3, 6) and (-, -6). Use of the graph is optional. 5

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27 Circles & Their Graphs 1.) Write an equation of a circle with center at (0, -8) and radius 3..) Which is an equation of the circle whose center is (, -3) and whose radius is 4? (1) ( x ) ( y 3) 16 () ( x ) ( y 3) 4 (3) ( x ) ( y 3) 4 (4) ( x ) ( y 3) 16 3.) Write an equation that represents the circle shown in the accompanying graph. \ 4.) Write an equation of a circle with center at (-, 0) and radius ) What is an equation of a circle with center at (-4, 1) and radius 3 3? (1) ( x 4) ( y 1) 7 () ( x 4) ( y 1) 7 (3) ( x 4) ( y 1) 3 3 (4) ( x 4) y 7 6.) If ( x 3) ( y 5) 9 is the equation of a circle, find the coordinates of center and the length of the radius. 7

28 7.) The center of a circle represented by the equation ( x ) ( y 3) 100 is located in quadrant (1) I () II (3) III (4) IV 8.) Write an equation of the circle whose center is (1, 3) and whose radius is. 9.) What are the coordinates of the center of the circle whose equation is ( x 3) ( y 7) 54? 10.) Write an equation of the circle with center at the origin and radius of ) Find the radius and the coordinates of the center of the circle whose equation is ( x 5) ( y 7) ) Find the radius and the coordinates of the center of the circle whose equation is ( x 10) ( y 1) 1 13.) What is the length of the radius of the circle whose equation is ( x 3) ( y 7) 54? 14.) Write an equation of the circle whose diameter has endpoints of (-, ) and (0, 4). Hint: Find center and radius! 15.) Write an equation of the circle whose diameter has endpoints of (5, 4) and (-3, ). See hint in #14! 16.) Write an equation of a circle with center at (-5, ) and passing through the point (1, ). See hint #14! 17.) Write an equation of a circle with center at (-3, -5) and passing through the point (0, -5). See hint #14! 8

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30 Circles, Standard Form, & Completing the Square 30

31 31 Use completing the square to put each equation in center-radius form. Name the center and radius of your circle y x y x y x y x y x y x x y x y x y x y x y x

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33 Solve Systems of Equations Graphically (1) () (3) 33

34 (4) (5) (6) 34

35 (7) (8) 35

36 (9) (10) 36

37 (11) Circle the system below that has exactly ONE solution. (1) Solve the given system of a circle and a line either algebraically or by finding the intersection points: y 3x 5 x y 5 37

38 Practice: Use the intersect feature on your graphing calculator to find all solution(s) to each system of equations. (13) x y 15 ( x ) ( y 1) 5 (14) x y 8 ( x 3) ( y 1) 16 (15) 3y x 6 ( x 3) ( y ) 9 (16) Solve the system of two circles by finding all intersection points: 38

39 Solve the following systems of equations graphically. (17) (18) ( x 1) ( x ) y y

40 (19) The epicenter of an earthquake is the point on the Earth s surface that is directly above the earthquake s origin. A seismograph can measure the distance to the epicenter, but not the direction to the epicenter. To locate the epicenter, readings from three seismographs in different locations are needed. a. For each seismograph location, sketch all possible locations on the graph for the epicenter. Seismograph readings: From location A (-5,6) the epicenter is exactly13 miles away. From location B (6,) the epicenter is exactly 10 miles away. From location C (0,0) the epicenter is exactly 6 miles away. b. Write the equations of the three circles sketched in part a. Seismograph readings: From location A (-5,6) the epicenter is 13 miles away. Equation: From location B (6,) the epicenter is 10 miles away. From location C (0,0) the epicenter is 6 miles away. c. State the coordinates of the epicenter of the earthquake. (, ) d. People could feel the earthquake up to 14 miles away. If you lived at (-3, 0), would you be able to feel the earthquake? Explain your reasoning mathematically. 40

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