Coordinate Geometry Midpoint of the Line Joining Two Points Areas of Triangles Parallel and Non-Parallel Lines Perpendicular Lines

Coordinate Geometry Objectives Midpoint of the Line Joining Two Points In this Module, you will learn how to find the midpoint of a line segment and apply it to solve problems.

A line AB joins points (x 1, y 1 ) and (x 2, y 2 ). M (x, y) is the midpoint of AB. M is the point Coordinate Geometry

P, Q, R and S are coordinates of a parallelogram and M is the midpoint of PR. Find the coordinates of M and S and show that PQRS is a rhombus. M is also the midpoint of QS. Coordinate Geometry Example 1

3 points have coordinates A(–1, 6), B(3, 2) and C(–5, –4). Given that D and E are the midpoints of AB and AC respectively, calculate the midpoint and length of DE. Let M be the midpoint of DE. Coordinate Geometry Example 2

Let M be the midpoint of AC. If A(2, 0), B(p, – 2), C(–1, 1) and D(3, r) are the vertices of a parallelogram ABCD, calculate the values of p and r. M is also the midpoint of BD. Coordinate Geometry Example 3

Areas of Triangles In this section, you will learn how to find the areas of Triangles figures given their vertices. Coordinate Geometry Objectives

Coordinate Geometry Areas of Triangles

Find the area of a triangle with vertices A(–2, –1), B(2, –3) and C(4, 3). The vertices A, B and C follow an anticlockwise direction. Coordinate Geometry Example 4

Parallel and Non-Parallel Lines In this section, you will learn how to apply the conditions for the gradients of parallel lines to solve problems. Coordinate Geometry Objectives

Consider the straight line with equation y = m 1 x + c 1 that makes an angle of θ 1 with the positive x-axis. Translate the line parallel to the x-axis. The new line has equation y = m 2 x + c 2 and makes an angle of θ 2 with the positive x-axis. The lines are parallel to each other. The lines make the same angle with the x-axis. The lines have the same gradient. θ 1 = θ 2 m 1 = m 2 Coordinate Geometry

Find the equation of the line which passes through the point (–2, 3) and is parallel to the line 2x + 3y – 3 = 0. Rearrange in the form y = mx + c. m in y = mx + c is the gradient of the line. Coordinate Geometry Example 5

Coordinate Geometry Perpendicular Lines In this section, you will learn how to apply the conditions for the gradients of perpendicular lines to solve problems. Objectives

Consider the straight line with equation y = m 1 x + c 1 that makes an angle of θ 1 with the positive x-axis. Rotate the line clockwise through 90°. The new line has equation y = m 2 x + c 2 and makes an angle of θ 2 with the negative x-axis. Applies to any two perpendicular lines. Coordinate Geometry

Two points have coordinates A(–2, 3) and B(4, 15). Find the equation of the perpendicular bisector of AB. Hence calculate the coordinates of the point P on the line 3y = x + 1 if P is equidistant from A and B. Find P. Solve simultaneou s equations Substitute for y Adding the equations Coordinate Geometry Example 6

The points A and B have coordinates (5, 2) and (3, 6) respectively. P and Q are points on the x-axis and y-axis and both P and Q are equidistant from A and B. (a)Find the equation of the perpendicular bisector of AB. (b) Find the coordinates of P and Q. Coordinate Geometry Example 7