Coordinate Geometry KUS objectives BAT find distances between
Coordinate Geometry KUS objectives BAT find distances between points BAT explore equations of lines, know the general equation of a line BAT use a new formula to find equations of lines Starter: state the gradient and y intercept of each line
Notes 1 Find the distance between these points: A (3, 7) and B (9, 19) 12 6
Notes 2 Find the gradient between these points: A (-1, 7) and B (19, 22) 15 20
Notes 3 Formula Find the gradient between these points: A (x 1, y 1) and B (x 2, y 2) y 1 – y 2 x 1 – x 2 Generally: Can you establish this result in a proof with clear steps?
Practice 1 Find the distance between these points Find the gradient between these points: A (4, 9) and B (7, 14) A (-2, -5) and B (-7, 7) A (8, -1) and B (7, -11) A (14, 3) and B (22, 13) A (3, 18) and B (9, 2) A (-7, -6) and B (14, -6)
Objectives BAT find distances between points BAT explore equations of lines, know the general equation of a line BAT use a new formula to find equations of lines
Notes 4 (x 2, y 2) What is gradient? Change in y Change in x y 2 - y 1 y 2 -y 1 x 2 -x 1 (x 1, y 1) x 2 - x 1 Find the gradient between these points (4, 9) and (7, 12)
notes We can generalise (x, y) y - y 1 x - x 1 Rearrange to: How can we use this? y- y 1 (x 1, y 1) x- x 1
WB 5 Find the line that joins these points (-2, 8) and (3, -7) Use this point (3, -7) x 1 is 3 and y 1 is -7 y – (-7) = -3 (x – 3) y + 7 = -3 x + 9 y = -3 x + 2
Practice 2 Find the equation of the line that: Joins (3, 7) and (9, 19) Joins (4, 2) and (7, -4) Goes through (3, 7) and has m = 3 Goes through (2, -3) and has m = -4 Joins (-1, -1) and (4, 14) Joins (8, 1) and (18, 6) Goes through (7, 3) and parallel to 3 y = x+7 Goes through (5, -5) and parallel to x+ 2 y =11
y WB 6 Find the equations of the lines that join these points (-6, 1) (2, 5) (-3, -5) activity: Geogebra three points x
Challenge y Choose three points of your own Work out the general equations of each line x
Objectives • 1 find distances between points • 2 explore equations of lines, know the general equation of a line • 3 use a new formula to find equations of lines
Notes The general equation of a line: ax + by + c = 0 y = 4 x + 6 4 x – y + 6 = 0 y= ½ x + 2 x – 2 y + 4 = 0 y= - x - 3 x+y+ 3 =0 y = -5 x + 1 5 x + y – 1 = 0
WB 7 Find the line that joins points (4, 9) and (8, 12) in the form ax + by + c = 0 Use this point (4, 9) x 1 is 4 and y 1 is 9 y – 9 = ¾ (x – 4) y– 9 = ¾ x – 3 y= ¾ x + 6 or 3 x - 4 y + 24 = 0
WB 8 Find the line that joins points (-2, 8) and (3, -7) in the form ax + by + c = 0 Use this point (3, -7) x 1 is 3 and y 1 is -7 y – (-7) = -3 (x – 3) y + 7 = -3 x + 9 y = -3 x + 2 or 3 x + y - 2 = 0
WB 9 Find the general equation of the line through (3, 7) that is perpendicular to y = 2 x + 8 Use gradient Use the point (3, 7) x 1 is 3 and y 1 is 7 y – 7 = -½ (x – 3) y - 7 = -½ x + 3/2 y = -½x + 17/ 2 or x + 2 y - 17 = 0
WB 10 Use gradient Use the point (3, 7) x 1 is 3 and y 1 is 7 y – 7 = -½ (x – 3) y - 7 = -½ x + 3/2 y = -½x + 17/ 2 or x + 2 y = 17
WB 11 The line l 1 has gradient -3 goes through (-2, 3) Line l 2 is perpendicular to l 1 and goes through (-2, 3) Find the equations of lines l 1 and L 2 y – 3 = -3 (x – (-2)) l 1 is y = -3 x - 3 y – 3 = 1/3(x – (-2)) l 2 is 3 y = x + 11
Practice Find the equation of the line in the form ax + by +c = 0 Perpendicular to y = 3 x + 2 And goes through (6, 12) Perpendicular to y = - 1 /2 x + 1 And goes through (-3, 8) Find the equation of the line perpendicular to: The midpoint of the line that joins (4, 9) and (10, 21) The midpoint of the line that joins (-3, -6) and (9, 2)
WB 12 Gradient is -2
WB 13
Summary You should be able to: • Rearrange equations of lines • Use to find the equation of a line from two points or gradient plus a point • Use to find the equation of a perpendicular line given enough information
KUS objectives BAT solve linear geometry problems self-assess One thing learned is – One thing to improve is –
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