IB Studies Topic 5.1 and 6.2: straight line and linear models. 5.1 Gradient, equation of a line, parallel and perpendicular lines, points of intersection of lines, graphs of linear functions. Coordinate Geometry

Assumed (prior) knowledge: MMidpoint of a line segment. DDistance between two points. Gradient, horizontal & vertical lines, parallel & perpendicular lines, equation of a straight line.

Gradient formula. Example: Find the gradient of the line joining the points (3,7) and (-5,6)

Prior knowledge formulas:  Distance between two points (x 1, y 1 ) and (x 2, y 2 ):  Coordinates of a midpoint of a line segment with endpoints (x 1, y 1 ) and (x 2, y 2 ):

 5.1 Equation of a line in two dimensions. The forms: y=mx+c and ax+by+d=0. Gradient, intercepts. Points of intersection of lines. Lines with gradients m 1 and m 2. Parallel lines m 1 = m 2. Perpendicular lines

Points A and Bgradientmidpointdistance A(3,5); B(4,10) A(-2,7); B(3,1) A(3,8); B(-5,1) What do you remember about the coordinate geometry? Which lines have zero gradient? Which lines have undefined gradient? horizontal lines vertical lines

Coordinate Geometry Distance, Midpoint, Gradient. YOU NEED TO KOWN WHERE TO FIND THE FORMULAE IN YOUR FORMULAS BOOKLET! 0 x1x1 x2x2 y2y2 y1y1 P(x 1,y 1 ) Q(x 2,y 2 ) Y-axis X-axis

Distance between 2 Points P(x 1, y 1 ) and Q(x 2, y 2 ) Example: Find the distance between A(1,-3) and B(5,7). Helpful Hints: 1)It is usually a good idea to make a little sketch. 2)Write down ALL point formulae used in coordinate geometry. 3)Label points clearly (x 1, y 1 ) & (x 2,y 2 ) 4)Is your answer what you expect or realistic for the problem. A(1,-3) B(5,7) ) Formula : (x 1, y 1 ) (x 2, y 2 ) 2) Show substitution into formula clearly. Use calculator to evaluate (5-1) 2 + (7--3) 2 The distance between A(1, -3) and B(5, 7) is

Mid-Point of two Points P(x 1, y 1 ) and Q(x 2, y 2 ) Example: Find the Mid-Point of A(1,-3) and B(5,7). Helpful Hints: 1)It is usually a good idea to make a little sketch. 2)Write down ALL point formulae used in coordinate geometry. 3)Label points clearly (x 1, y 1 ) & (x 2,y 2 ) 4)Is your answer what you expect or realistic for the problem. A(1,-3) B(5,7) ) Formula: (average point / co-ordinates) 2) Show substitution into formula clearly. The mid-point of A(1, -3) and B(5, 7) is (3, 2). (x 1, y 1 ) (x 2, y 2 )

Example: Find the gradient, m, of the line passing through points P(-3,5) and Q(5,-5). Helpful Hints: 1)It is usually a good idea to make a little sketch. 2)Write down ALL point formulae used in coordinate geometry. 3)Label points clearly (x 1, y 1 ) & (x 2,y 2 ) 4)Is your answer what you expect or realistic for the problem. (x 1, y 1 )(x 2, y 2 ) 1) Formula : 2) Show substitution into formula clearly. The gradient of the line passing through P(- 3,5) and Q(5,-5) is : P(-3, 5) Q(5,-5) 10 8 

Using your calculator ›F›Find the gradient between (-1,3) and (5,-6)

Exercise 1) Given the two points A(-3,7) and B(5,1) find the a) distance AB b) point C, the mid-point of interval AB c) gradient, m AB, of the line joining AB Helpful Hints: 1)It is usually a good idea to make a little sketch. 2)Write down ALL point formulae used in coordinate geometry. 3)Label points clearly (x 1, y 1 ) & (x 2,y 2 ) 4)Is your answer what you expect or realistic for the problem. A(-3,7) B(5,1) (x 1, y 1 ) (x 2, y 2 ) The distance AB is d = 5.66 The mid-point C is (1,4) The gradient of line AB is

Example: Given that M(1,-3) is a midpoint of line segment from A(2,7) to B, find the coordinates of point B. Using the midpoint formula, we can form two equations Use your calculator to solve: Therefore B(0,-13)

Exercise: 2) Given the two points P(-5,-3) and Q(6,4) find the a) distance PQ b) point R, the mid-point of interval PQ c) gradient, m PQ, of the line joining PQ Helpful Hints: 1)It is usually a good idea to make a little sketch. 2)Write down ALL point formulae used in coordinate geometry. 3)Label points clearly (x 1, y 1 ) & (x 2,y 2 ) 4)Is your answer what you expect or realistic for the problem. P(-5,-3) Q(6,4) (x 1, y 1 ) (x 2, y 2 ) The distance PQ is d = 13.0 The mid-point R is The gradient of line PQ is

Exercise: 3) Given the two points L(8,-3) and M(-1,5) find the a) distance LM b) point K, the mid-point of interval LM c) gradient, m LM, of the line joining LM Helpful Hints: 1)It is usually a good idea to make a little sketch. 2)Write down ALL point formulae used in coordinate geometry. 3)Label points clearly (x 1, y 1 ) & (x 2,y 2 ) 4)Is your answer what you expect or realistic for the problem. M(-1,5) L(8,-3) (x 1, y 1 )(x 2, y 2 ) The distance LM is d = 12.0 The mid-point K is The gradient of line LM is