.(3, 2) Co-ordinate Geometry Co-ordinates. Every point has two co-ordinates. Plot the following points on the plane. A (4, 1) D (2, 5) G (6, 3)

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1 Co-ordinate Geometry Co-ordinates Every point has two co-ordinates. (3, 2) x co-ordinate y co-ordinate Plot the following points on the plane..(3, 2) A (4, 1) D (2, 5) G (6, 3) B (3, 3) E ( 4, 4) H (6, 3) C ( 6, 5) F ( 1, 2) I ( 3, 1.5) Maths Excel Class

2 Co-ordinate Geometry Gradient The gradient (m) is the slope of the line. Positive gradient (m > 0) Negative gradient (m < 0) m = 0 A bigger gradient makes a steeper line. m = 2 m = 1 m = 1 m = 2 Maths Excel Class

3 Co-ordinate Geometry Gradient = : ; 8 9; : Example 1 Find the gradient of the following line..(4, 3) m = 5 6 Rise = 3 Run = 4 Example 2 Find the gradient of the line. (3, 2).. (6, 7) Rise = 7 2 = 5 m = : ; 8 9; : = <9= >95 =? = Run = 6 3 = 3 Maths Excel Class

4 Co-ordinate Geometry Gradient Questions Plot the following points and find the gradient. 1. (5, 3) and (2, 1) 4. (5, 4) and ( 2, 1) 2. (7, 5) and (1, 1) 5. ( 3, 4) and (9, 1) 3. (3, 8) and (0, 0) 6. ( 4, 1) and ( 8, 2) Find the gradient of the line for each pair of points. 1. (5, 8) and (1, 4) 2. ( 3, 5) and (10, 2) 3. (10, 35) and (2, 3) 4. ( 4, 7) and ( 5, 1) 5. (22, 7) and (2, 2) Maths Excel Class

5 Co-ordinate Geometry Equation of a line There are two forms for the equation of a line. General form: ax + by + c = 0 Gradient Intercept form: y = mx + b gradient y-intercept There are three ways to find the equation of a line depending on what information you have. 1. Gradient and point y y 1 = m(x x 1 ) 2. Two points y y 1 x x 1 = y 2 y 1 x 2 x 1 3. Gradient and y-intercept y = mx + b Testing if a point is on the line Sub the point into the equation. If it satisfies the equation, the point lies on the line. Example Does the point (3, 4) lie on the line 2x 3y + 6 = 0? LHS = y = = 0 = RHS x The point (3, 4) is on the line. Maths Excel Class

6 Co-ordinate Geometry Equation of a line x- intercepts and y-intercepts y 3 4 x The y-intercept is 3. The co-ordinates are (0, 3). The x-intercept is 4. The co-ordinates are ( 4, 0). Example 1 The equation of a line is 3x 4y + 12 = 0. Find the x- intercept and y-intercept. y-intercept Sub x = 0 into the equation. 3x 4y + 12 = 0 3(0) 4y + 12 = 0 4y = 12 y = 3 The y-intercept is 3. x-intercept Sub y = 0 into the equation. 3x 4y + 12 = 0 3x 4(0) + 12 = 0 3x = 12 x = 4 The x-intercept is 4. Maths Excel Class

7 Co-ordinate Geometry Equation of a line Find the x-intercept and y-intercept of the following lines. Sketch the graphs. 1. 5x 2y + 10 = x 2y + 6 = x y + 10 = y 7x + 14 = 0 5. y 2x + 6 = 0 6. y = T = x y = = x y = 3x 11 Maths Excel Class

8 Co-ordinate Geometry Equation of a line Find the equation of the following lines in gradient-intercept form. 1. The line has a gradient of 2 and passes through the point (3, 4). 2. The line has a gradient of 7 and passes through the point ( 5, 2). 3. The line has a gradient of T and passes through the point (1, 4). = 4. The line passes through the points (1, 3) and (6, 8). 5. The line passes through the points (6, 2) and (9, 3). 6. The line has a gradient of 5 and a y-intercept of 4.? Maths Excel Class

9 Co-ordinate Geometry Equation of a line Find the equation of the following lines in general form. 1. The line has a gradient of? and passes through the point (1, 2). > 2. The line has a gradient of = and passes through the point (5, 3). U 3. The line passes through the points (3, 1) and (5, 6). 4. The line passes through the points (2, 11) and ( 3, 1). 5. The line has a gradient of T and an x-intercept of 8. 6 Maths Excel Class

10 Co-ordinate Geometry Equation of a line 1. Show that the point (0, 2) lies on the line x 4y + 8 = Show that the point ( 2, 4) lies on the line y = 8x Show that the point (1, 7) lies on the line y = 4x 11. Maths Excel Class

11 Co-ordinate Geometry Sketching For the equation y = 2x + 3 fill in the table of values. y x Sketch the graph. Remember: Always label the x and y axes with both letter and arrow Always put an arrow at each end of the line sketched Label the x-intercept and y-intercept if possible To sketch a graph, you only need to know two points. A table of values is not necessary. Changing the equation into gradient-intercept form is useful. Maths Excel Class

12 Co-ordinate Geometry Sketching Example 1 Sketch the line that passes through the points (1, 4) and (5, 6). Plot the points... (5, 6).. (5, 6) (1, 4) (1, 4) Example 2 Sketch the graph of 2x 3y + 6 = 0. x-intercept: 3 2 y-intercept: 2 3 Example 3 Sketch the graph of y = 3 x y-intercept: 2 = 5 6. (4, 5). (4, 5) 2 run = 4 rise = 3 2 Maths Excel Class

13 Co-ordinate Geometry Sketching 1. Sketch the line that passes through the points (3, 5) and (9, 7). What is the gradient? 2. Sketch the line that passes through the points ( 6, 5) and (2, 5). What is the gradient? 3. Sketch the line that passes through the points (1, 8) and (10, 1). What is the gradient? Maths Excel Class

14 Co-ordinate Geometry Sketching 4. Graph the line with gradient = and a y-intercept of Graph the line with gradient 5 and a y-intercept of Graph the line with gradient ] and a y-intercept of 3. U Maths Excel Class

15 Co-ordinate Geometry Sketching 7. Sketch the line with equation y = 1 x Graph the linear function y = 2x Graph the linear function y = 3x 4. Maths Excel Class

16 Co-ordinate Geometry Sketching 10. Sketch the line with equation 3x 2y 12 = Graph the linear function 8x + 3y 24 = Graph the linear function 4x + 3y + 12 = 0. Maths Excel Class

17 Co-ordinate Geometry Parallel lines Parallel lines: Gradients are the same m 1 = m 2 Example 1 Is Line 1 parallel to Line 2? Line 1 Line 2. (2, 6). (4, 5) 1 2 Line 1 passes through Line 2 passes through (2, 6) and (0, 1). (4, 5) and (2, 0). m T = =? = m = = =? = m T = m = Line 1 is parallel to Line 2. Example 2 Are the lines parallel? Line 1: y = 1 x + 6. Line 2: x 4y + 23 = 0 4 m 1 = 1 4 4y = x + 23 y = 1 23 x m 2 = 1 4 m 1 = m 2 The lines are parallel. Maths Excel Class

18 Co-ordinate Geometry Parallel lines 1. Line 1 passes through the points (4, 5) and (2, 2). Line 2 passes through the points (8, 7) and (6, 4). Sketch both lines on the same plane and prove they are parallel. 2. Line A has the equation x 7y 33 = 0. Line B is parallel to Line A and passes through the point (5, 1). Find the equation of Line B and sketch both lines on the same plane. 3. Prove the following lines are parallel and sketch each pair on the same plane. a. Line 1: y = 4 5 x + 1 Line 2: 4x 5y + 20 = 0 b. Line 1: 7x + 3y + 21 = 0 Line 2: 7x + 3y 42 = 0 Maths Excel Class

19 Co-ordinate Geometry Perpendicular lines Perpendicular lines: Gradients are the negative reciprocal of each other m 1 = 1 m 2 or m 1 m 2 = 1 Perpendicular lines intersect at an angle of 90. AB CD. A D C B Example 1 Is Line 1 perpendicular to Line 2? Line 1: 4x 3y + 3 = Line 2: 3x + 4y 16 = 0 Line 1 has equation Line 2 has equation 4x 3y + 3 = 0. 3x + 4y 16 = 0. Change to G-I form. y = 4 3 x + 1 y = 3 4 x + 4 m T = 4 3 m = = 3 4 m 1 m 2 = 4 3 = 1 Line 1 is perpendicular to Line Maths Excel Class

20 Co-ordinate Geometry Perpendicular lines 1. Write the negative reciprocal of the following: a. = 5 c. 3 b. > < d Line 1 passes through the points (3, 7) and (1, 4). Line 2 passes through the points (3, 3) and (0, 5). Prove that Line 1 is perpendicular to Line 2 and sketch both lines on the same plane. 3. Line 1 passes through the points (6, 9) and (5, 3). Line 2 passes through the points (8, 4) and (2, 5). Prove that Line 1 is perpendicular to Line 2 and sketch both lines on the same plane. 4. Line A has the equation 2x 5y + 1 = 0. Line B is perpendicular to Line A and passes through the point (3, 4). Find the equation of Line B and sketch both lines on the same plane. Maths Excel Class

21 Co-ordinate Geometry Midpoint Midpoint: find the average of each co-ordinate. g x T + x = 2, y T + y = h 2 x co-ordinate = ; :f; 8 = y co-ordinate = 7 :f7 8 = Example 1 Find the midpoint between (1, 2) and (5, 8).. (5, 8). (1, 2) Midpoint: x co-ordinate = Tf? = y co-ordinate = =f] = = 3 = 5 The midpoint is (3, 5).. (5, 8). (3, 5). (1, 2) Maths Excel Class

22 Co-ordinate Geometry Midpoint Plot the following points and find the midpoint. 1. (3, 4) and (7, 8) 2. 2, 6 and 4, (5, 1) and (1, 9) 4. Point A is the y-intercept of the line y = 2x + 3. Point B has co-ordinates 4, 11. a. Prove that Point B lies on the line y = 2x + 3. b. Sketch the line y = 2x + 3 indicating Point A and Point B. Find the midpoint of A and B and add it to the diagram. Maths Excel Class

23 Co-ordinate Geometry Distance Distance = k(x 2 x 1 ) 2 + (y 2 y 1 ) 2 units Example Find the distance (D) between the points (2, 4) and (6, 7).. (2, 4) D. (6, 7) D is the hypotenuse of a right-angled triangle. The height of the triangle is (7 4). The base of the triangle is (6 2). (2, 4). (6, 7). Base = 6 2 D Height = 7 4 Using Pythagoras theorem: D = = 6 2 = = D = 6 2 = = D = 4 = + 3 = = = 25 D = 5 units Maths Excel Class

24 Co-ordinate Geometry Distance 1. Point A has co-ordinates 2, 7 and Point B has co-ordinates 8, 15. Plot the points and find the distance of line AB. 2. Point A has co-ordinates 4, 1 and Point B has co-ordinates 8, 7. Plot the points and find the distance of line AB. 3. Point A has co-ordinates 11, 10 and Point B has co-ordinates 6, 2. Plot the points and find the distance of line AB. 4. Point A has co-ordinates 9, 7 and Point B has co-ordinates 2, 6. Plot the points and find the distance of line AB. Maths Excel Class

25 Co-ordinate Geometry Distance 5. Point A has co-ordinates 4, 0 and Point B has co-ordinates 6, 10. Plot the points and find the distance of line AB. 6. Point A has co-ordinates 2, 2 and Point B has co-ordinates 3, 1. Plot the points and find the distance of line AB. 7. On a map, a red flag is placed at co-ordinates 5, 8 and a green flag is placed at co-ordinates 8, 1. Plot the position of the two flags and find the difference between them. 8. Point A is the midpoint between 3, 8 and 5, 4. Point B has coordinates 13, 12. What is the distance of line AB? Maths Excel Class

26 Co-ordinate Geometry Distance 9. Plot point A 3, 2, point B 5, 2 and point C 5, 8. a. Find the distance of AB. b. Find the distance of BC. c. Using either the distance formula or Pythagoras theorem, find the distance of AC. d. Find the area of ABC. e. Find the equation of line AC. f. Find the equations of lines AB and BC. Maths Excel Class

27 Co-ordinate Geometry Perpendicular Distance Perpendicular distance = ax 1f by 1 f c ka 2 f b 2 units Perpendicular distance is the shortest distance between a point and a line. Example 1 Find the perpendicular distance between the point 4, 1 and the line 2x 3y + 6 = 0. 2x 3y + 6 = 0 D.(4, 1) Distance = ax 1 + by 1 + c a 2 + b 2 From the equation 2x 3y + 6 = 0. a = 2 b = 3 From the point 4, 1 x T = 4 c = 6 y T = 1 Distance = ( 3) 2 = = = units = units (rational denominator) Maths Excel Class

28 Co-ordinate Geometry Perpendicular Distance Draw in the perpendicular distance between the following lines and points..... REMEMBER: To use the perpendicular distance formula, the equation of the line must be in general form. Arrange the equation so that the x comes before the y. Maths Excel Class

29 Co-ordinate Geometry Perpendicular Distance 1. Find the perpendicular distance between the point 1, 8 and the line 4x + 3y + 1 = Find the perpendicular distance between the point 3, 2 and the line 8x + 6y 5 = Find the perpendicular distance between the point 5, 1 and the line x 7y 10 = Find the perpendicular distance between the point 3, 5 and the line 2y + 7x + 4 = 0. Maths Excel Class

30 Co-ordinate Geometry Perpendicular Distance 5. Find the perpendicular distance between the point 4, 2 and the line y = 3 x Sketch the line y = 1 x 2 and find the perpendicular distance from the 4 point 4, 2 to the line. 7. Sketch the line y = 4 3 x 11 and find the perpendicular distance from the point 1, 2 to the line. Maths Excel Class

Section Graphs and Lines

Section Graphs and Lines Section 1.1 - Graphs and Lines The first chapter of this text is a review of College Algebra skills that you will need as you move through the course. This is a review, so you should have some familiarity