Outcomes
A student:

 MA5.31WM
uses and interprets formal definitions and generalisations when explaining solutions and/or conjectures

 MA5.32WM
generalises mathematical ideas and techniques to analyse and solve problems efficiently

 MA5.33WM
uses deductive reasoning in presenting arguments and formal proofs

 MA5.38NA
uses formulas to find midpoint, gradient and distance on the Cartesian plane, and applies standard forms of the equation of a straight line
Content
 Students:
 Find the midpoint and gradient of a line segment (interval) on the Cartesian plane (ACMNA294)
 use the concept of an average to establish the formula for the midpoint, M, of the interval joining two points \( (x_1,y_1) \) and \( (x_2,y_2) \) on the Cartesian plane: \( M(x,y) = \left( \frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right) \)
 explain the meaning of each of the pronumerals in the formula for midpoint (Communicating)
 use the formula to find the midpoint of the interval joining two points on the Cartesian plane
 use the relationship \( \textrm{gradient} = \dfrac{\textrm{rise}}{\textrm{run}} \) to establish the formula for the gradient, m, of the interval joining two points \( (x_1,y_1) \) and \( (x_2, y_2) \) on the Cartesian plane: \( m = \frac{y_2  y_1}{x_2  x_1} \)
 use the formula to find the gradient of the interval joining two points on the Cartesian plane
 explain why the formula \( m = \frac{y_1  y_2}{x_1  x_2} \) gives the same value for the gradient as \( m = \frac{y_2  y_1}{x_2  x_1} \) (Communicating, Reasoning)
 Find the distance between two points located on the Cartesian plane (ACMNA214)
 use Pythagoras' theorem to establish the formula for the distance, d, between two points \( (x_1,y_1) \) and \( (x_2,y_2) \) on the Cartesian plane: \( d = \sqrt{(x_2x_1)^2 + (y_2y_1)^2} \)
 explain the meaning of each of the pronumerals in the formula for distance (Communicating)
 use the formula to find the distance between two points on the Cartesian plane
 explain why the formula \( d = \sqrt{(x_1x_2)^2 + (y_1y_2)^2} \) gives the same value for the distance as \( d = \sqrt{(x_2x_1)^2 + (y_2y_1)^2} \) (Communicating, Reasoning)
 Sketch linear graphs using the coordinates of two points (ACMNA215)
 sketch the graph of a line by using its equation to find the x and yintercepts
 Solve problems using various standard forms of the equation of a straight line
 describe the equation of a line as the relationship between the x and ycoordinates of any point on the line
 recognise from a list of equations those that can be represented as straightline graphs (Communicating, Reasoning)
 rearrange linear equations in gradientintercept form (\(y=mx+b\)) into general form \(ax+by+c=0\)
 find the equation of a line passing through a point \( (x_1,y_1) \), with a given gradient m, using:
 pointgradient form: \(y  y_1 = m(x  x_1) \)
 gradientintercept form: \(y = mx + b \)
 find the equation of a line passing through two points
 recognise and find the equation of a line in general form \(ax+by+c=0\)
 Solve problems involving parallel and perpendicular lines (ACMNA238)
 find the equation of a line that is parallel or perpendicular to a given line
 determine whether two given lines are perpendicular
 use gradients to show that two given lines are perpendicular (Communicating, Problem Solving)
 solve a variety of problems by applying coordinate geometry formulas
 derive the formula for the distance between two points (Reasoning)
 show that three given points are collinear (Communicating, Reasoning)
 use coordinate geometry to investigate and describe the properties of triangles and quadrilaterals (Communicating, Problem Solving, Reasoning)
 use coordinate geometry to investigate the intersection of the perpendicular bisectors of the sides of acuteangled triangles (Problem Solving, Reasoning)
 show that four specified points form the vertices of particular quadrilaterals (Communicating, Problem Solving, Reasoning)
 prove that a particular triangle drawn on the Cartesian plane is rightangled (Communicating, Reasoning)