Analytical geometry Multiple choice questions 1. Temperature readings on any given day in Québec can vary greatly. The temperatures for a fall day in Montreal were recorded over a 10-hour interval. The graph below represents this situation, with x being the number of hours that have elapsed since the temperature was first recorded, and y being the temperature in degrees Celsius. What are the values of the slope, the zero, and the initial value, respectively? A), 5, -8 C), -8, 5 B), -8, 5 D), 5, -8
2. The two main streets of a city are perpendicular. These can be drawn in a Cartesian plane. The line representing street A passes through the points (0, -60) and (70, 10). The line representing street B passes through the point (90, -30). Which of the following equations defines the line representing street B? A) y = - x 60 C) y = x + 60 B) y = x 60 D) y = - x + 60 3. What is the equation of the line that passes through point (9, 7) and is perpendicular to the line y = x 5? A) y = - x + 10 C) y = -3 x + 34 B) y = -3 x + 7 D) y = 3 x 20
4. While out walking, Ben and Gabriella stopped abruptly when a skunk crossed their path. They ran in opposite directions, each running the same distance before coming to a stop. On an imaginary Cartesian plane, Ben would be stopped at the point (23, 35) and Gabriella would be stopped at the point (-54, 17). What are the coordinates of the point at which they first would have seen the skunk? A) (-31, 52) C) (38.5, 9) B) (-15.5, 26) D) (38.5, 26)
5. The positions of 3 islands are given by the following points : F(-10, 4), Q(2, 8) and R(7, -7) What distance, to the nearest tenth, would a boat have to travel to make a round trip to the three islands? A) 39.7 C) 48.6 B) 44.9 D) 50.9
6. In the Cartesian plane, points R(-16, 45) and Q(26, 5) are the endpoints of one of the diameters of a circle. Which of the following statements is true? A) The radius of this circle is 20.6 units. B) The radius of this circle is 29 units. C) The coordinates of the centre of this circle are (5, 20). D) The coordinates of the centre of this circle are (21, 25).
Short answer questions 7. Points P(624, 36) and Q(956, 86) are the endpoints of one of the diameters of a circle. What are the coordinates of the centre of this circle? 8. The rule of a function f is f(x) = 5 x 17. What is the zero of this function? 9. In the Cartesian plane below, segment AM is a median of triangle ABC. What are the coordinates of point M?
10. Point A is the intersection of lines 1 and 2 represented in the Cartesian plane on the right. The equation of line 1 is 5 x + 2 y + 156 = 0 The equation of line 2 is y = 6 x 183. What are the coordinates of point A?
11. The city of Chateauguay uses a Cartesian grid for mapping out roads. Elm Street has endpoints (-1, 8) and (3, -4). The town manager wishes to find the equation of a line representing Valour Lane which is perpendicular to Elm Street and passes through the point (6, 5). What is the equation of the line that represents Valour Lane?
12. Find the slope and the x -intercept of the line represented by the following equation: 3 x 4 y + 24 = 0 13 Mrs. Perez had a triangular-shaped pool installed in her back yard. A diving board was placed at corner A. The company installed a rope connecting the midpoints of sides AB and AC to designate the diving area. This situation is illustrated in the Cartesian plane below, which is scaled in metres. Show that the rope is parallel to side BC of the pool.
14. Perpendicular lines and are drawn in the Cartesian plane below. What is the x intercept of lines?
15. Luke wants to cut a piece of metal into the shape of a trapezoid. To draw the outline of this piece, he uses a software program that shows only the first quadrant of the Cartesian plane, scaled in centimetres. The metal piece is represented by the region between the axes the two parallel lines of the trapezoid. The equation of the first parallel line, shown in the Cartesian plane on the right, is f ( x ) = + 12. The y -intercept of the second parallel line is 3. What is the perimeter of the piece of metal? Show all your work.
.16 The water main in a new residential development must be made longer. The town surveyor drew the new part of the water main on a Cartesian plane, where represents the existing water main M is the midpoint of and represent the new water main Rounded to the nearest tenth, what is the total length of the new water main, FGM? Show all your work.
17. A municipal land surveyor drew a graph on a Cartesian plane showing two sections of a planned water main. These two sections are represented by lines AB and CD whose equations are as follows : Line AB : y = 2 x + 2 Line CD : y = - x + 5 These lines meet at point E. A valve is located at points E and F. What is the distance between the valve at point E and the valve at point F? Round off your answer to the nearest tenth. Show your work.
18. Rosalie spent a week at the Beausoleil Campground. She drew a map of the campground, with the y -axis passing through the games area (G) and the showers (S). She located her campsite (C) and the swimming pool (P). She also indicated the coordinates of some points. The games area, her campsite, and the swimming pool lie in a straight line. The equation of the line that passes through the showers and Rosalie's campsite is y = 450 0.25 x. Map of Beausoleil Campground To the nearest metre, what is the distance between Rosalie's campsite and the showers? Show all your work.
19. A few neighbourhood streets are represented in the Cartesian plane below. The streets of this neighbourhood are linear. Nevil Street is parallel to Angus Street. The scale of this graph is in metres. What is the length of Angus Street to the nearest metre? Show all your work.
20. In the Cartesian plane below, lines PQ and RS intersect the y -axis at points P and R respectively. Lines PQ and RS intersect at point Q. The equation of line PQ is y = - x + 42. What is the area of triangle PQR? Show all your work.