C 4 up Page | 1 Olympic College - Topic 6 Cartesian Coordinates and Slopes Topic 6 Cartesian Coordinates and Slopes 1. Cartesian Coordinate System Definition: The “Cartesian Coordinate System” way of representing the position of a point in two dimensional space. It is constructed with two perpendicular axes the horizontal one is called the x-axis while the vertical one is called the y-axis. The two axes cross at the origin and we give the location of any point in terms of a number pair (called coordinates) written in the form (a,b) where a is the distance along the x-axis of the point and b is the distance along the y-axis of the point. By this means every point in two dimensional space can be represented by a unique coordinate. Example 1: Plot the points with coordinates A(4,3), B (-2,4), C(0,-3) Solution: Example 2: What are the coordinates of the points on the Cartesian graph below. Solution:A(-2,4) B(2,4) C(-3,0) D(4,2) E(-3,-3) F(0,-4) G(3,-3) x y to the right up 4 5 A(4,3) x y B(-2,4) to the left x y to the right C(0,-3) x y A B D E F G

Page | 2 Olympic College - Topic 6 Cartesian Coordinates and Slopes Exercise 1A: 1.Plot the points with coordinates A(5,1), B (-3,2), C(0,4), D(-2,-4), E (-2,0), F(4,3). 2.(a) What are the coordinates of the following points? (b) Which points have an x-coordinate of – 3? (c) Which points have a y-coordinate of – 3? (d) Which points have negative x-coordinates? (e) Which points have positive y-coordinates? (f) Which point(s) have both their x-coordinate and y-coordinate equal to each other? (g) Which point(s) have both their x-coordinate and y-coordinates negative? (h) Which point(s) have a y-coordinate that is two more than its x-coordinate? (i) Which point(s) have a y-coordinate that is two less than its x-coordinate? (k) Which point(s) have a y-coordinate that is four less than its x-coordinate? (l) Which point(s) have a y-coordinate that is half its x-coordinate? x y -4 A B C D E F G x On the coordinate plane below, which point represents the coordinates (4,-2) y A B C D

Run = 0 Rise = - 8 Rise = 8 Page | 3 Slope = Negative Slope:Zero Slope (horizontal line) Undefined Slope(vertical line) Slope ==== ==== ==== m=m=0m= undefined x Olympic College - Topic 6 Cartesian Coordinates and Slopes 2. Slope of a Line Segment The slope of a line segment is a number that measures how steep a line is. It is defined as the ratio between the vertical distance called the “Rise” and the horizontal distance called the “Run”. A graphical representation of a line segment with a positive slope is given below. y The other possible slopes of line segments are negative slopes, zero slopes and undefined slopes. An example of each type of slope is given below. Notice we typically use the letter m to represent the slope of a line segment. Example 1: 2 Rise = Run = y Run = x x y Rise = 0 Run = x y -4 -5

Page | 4 Olympic College - Topic 6 Cartesian Coordinates and Slopes It is easy to see when a line segment will have a positive, negative, zero or undefined slope by just looking at the graph. Example 2: (a) Which of the following line segments have positive slopes? (b) Which of the following line segments have negative slopes? (c) Which of the following line segments have a slope of zero? (d) Which of the following line segments have undefined slopes? Solution: (c) Lines which are horizontal will have a slope of zero. So line 8 has a slope of zero. Solution: (d) Lines which are vertical will have an undefined slope. So line 7 will have an undefined slope. x y x Solution: (a) Lines which go up (increase) as you move from left to right will have positive slopes. So lines 1,2 & 5 are positive slopes. y x (b) Lines which go down (decrease) as you move from right to left have negative slopes. So lines 3,4 & 6 have negative slopes. y x y 7 8

Page | 5 Slope = m = Solution: (a) The points are A(3,4) and B(7,9) so x 1 = 3, y 1 = 4, x 2 = 7 and y 2 = 10 Slope=m==== 3 (b) The points are C(-7,2) and D(3,– 4) so x 1 = - 7, y 1 = 2, x 2 = 3 and y 2 = - 4 Slope=m==== (c) The points are E(2,5) and F(7,5) so x 1 = 2, y 1 = 5, x 2 = 7 and y 2 = 5 Slope=m==== 0 (A horizontal line) (d) The points are G(2,-2) and H(2,9) so x 1 = 2, y 1 = – 2, x 2 = 2 and y 2 = 7 Slope=m==== undefined(A vertical line) Rise = y 2 – y 1 Run = x 2 – x 1 A(x 1,y 1 ) We only need the coordinates of the end points of the line segment in order to calculate its slope. This is useful as we do not need to plot the point or be given a Cartesian Graph. Example 3: (a) Find the slope of the line segment joining the points A(3,4) and B(7,9). (b) Find the slope of the line segment joining the points C(-7,2) and D(3,– 4). (c) Find the slope of the line segment joining the points E(2,5) and F(7,5). (d) Find the slope of the line segment joining the points G(2,-2) and H(2,9). B(x 2,y 2 ) x Olympic College - Topic 6 Cartesian Coordinates and Slopes We can also find the slope of a line segment by using the coordinates of the ends of the line. The slope of the line segment that joins the points A(x 1,y 1 ) and B(x 2,y 2 ) is given by the formula. y

Olympic College - Topic 6 Cartesian Coordinates and Slopes There are a number of useful facts that use slopes. Two of the most common are that if the slope of two line segments is equal then the line segments are parallel. The other fact is that if two line segments are perpendicular (cross at 90 0 ) then the product of their slopes will be – 1. This last fact only works if neither of the lines is horizontal or vertical. Example 4: Which of the following pairs of line segments are parallel and which are Solution:First find the coordinates of the end points of the line segments. A(2,4) and B(-4,-4) so x 1 = 2, y 1 = 4, x 2 = -2 and y 2 = -4 Slope = m AB = === C(-1,-4) and D(2,0) so x 1 = -1, y 1 = -4, x 2 = 2 and y 2 = 0 Slope = m CD = == Since the slopes of the two line segments are equal we can conclude that the two line segments are parallel. Solution:First find the coordinates of the end points of the line segments. E(-4,3) and F(4,-1) so x 1 = -3, y 1 = 3, x 2 = 4 and y 2 = -1 Slope = m EF = === G(1,-2) and H(3,2) so x 1 = 1, y 1 = -2, x 2 = 3 and y 2 = 2 Slope = m CD = ===2 Conclusion: Since the produce of the two slopes m EF * m CD = we can conclude that the two line segments are perpendicular. Page | 6 x perpendicular. y A B C D E F G H x y

y Page | 7 Olympic College - Topic 6 Cartesian Coordinates and Slopes Exercise 2A: 1. 2.(a)Which of the following line segments have positive slopes? 2.(b)Which of the following line segments have negative slopes? 2.(c) 2.(d) Which of the following line segments have a slope of zero? Which of the following line segments have undefined slopes? 3.(a) 3.(b) 3.(c) 3.(d) 3.(e) 3.(f) 3.(g) 3.(h) Find the slope of the line segment joining the points A(2,7) and B(4,15). Find the slope of the line segment joining the points C(–2,8) and D(2,– 4). Find the slope of the line segment joining the points E(–5, –2) and F(3,10). Find the slope of the line segment joining the points G(8, –1) and H(8,4). Find the slope of the line segment joining the points J(–3,5) and K(3,5). Find the slope of the line segment joining the points L(–5, –5) and M(15,– 10). Find the slope of the line segment joining the points N(1.2,0) and O(–0.8,0.5). Find the slope of the line segment joining the points P(, ) and Q(, ) x y Calculate the Slope of each line below, leaving your answer as a fraction in its simplest form where necessary. x

Olympic College - Topic 6 Cartesian Coordinates and Slopes 4. Which of the following pairs of line segments are parallel and which are perpendicular or neither? 5. Which of the following pairs of line segments are parallel and which are perpendicular or neither? 6. Is the line segment joining the points A(3,5) and B(7,7) parallel or perpendicular to the line segment joining the points C(– 1,2) and D(1,– 2)? 7. Is the line segment joining the points A(2,2) and B(0,4) parallel or perpendicular to the line segment joining the points C(– 3,0) and D(– 7,4)? 8. Is the line segment joining the points A(3,5) and B(3,7) parallel or perpendicular to the line segment joining the points C(– 1,2) and D(7,2)? 9. Is the line segment joining the points A(13,– 8 ) and B(3, – 3 ) parallel or perpendicular to the line segment joining the points C(– 4,3) and D(– 14,8)? Page | 8 A B C D x y E F G H x y K L I J x y x A CBCB D y E G HFHF x y LKLK JIJI x y

Page | (a) 2.(b) 2.(c) 2.(d) A(-2,4) B(2,4) C(-3,0) D(4,2) E(-3,-3) F(0,-4) G(3,-3) C and E E and G A,C and E 2.(e) 2.(f) A,B and D E 2.(g) 2.(h) EBEB 2.(i)D 2.(k) 2.(l) FDFD 3.B Exercise 2A: 1. m 1 =m 2 = 0m 3 =m 4 =m 5 = undefined 2.(a) 2.(c) 3.(a) 3.(c) 3.(e) 3.(g) Lines 2,4 and 5 Line 3 Slope = m AB = 4 Slope = m EF = Slope = m JK = Slope = m NO = – (b) 2.(d) 3.(b) 3.(d) 3.(f) 3.(h) Lines 1,7 and 8 Line 6 Slope = m CD = – 3 Slope = m GH undefined Slope = m LM = Slope = m PQ = AB and CD are perpendicular; EF and GH neither; IJ and KL are parallel. AB and CD are neither; EF and GH perpendicular; IJ and KL are parallel. perpendicular parallel perpendicular parallel Olympic College - Topic 6 Cartesian Coordinates and Slopes Solutions. y Exercise 1A: A x B C432C432 D E F G