Chapter 7 Coordinate Geometry 7.1 Midpoint of the Line Joining Two Points 7.2 Areas of Triangles and Quadrilaterals 7.3 Parallel and Non-Parallel Lines 7.4 Perpendicular Lines

Coordinate Geometry Objectives 7.1 Midpoint of the Line Joining Two Points In this lesson, 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. Construct a right angled triangle ABC. Construct the midpoints D and E of the line segments AC and BC. Take the mean of the coordinates at the endpoints. D is and E is M is the point Take the x-coordinate of D and the y-coordinate of E. 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 4

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 Exercise 7.1, qn 3

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 Exercise 7.1, qn 4

7.2 Areas of Triangles and Quadrilaterals In this lesson, you will learn how to find the areas of rectilinear figures given their vertices. Coordinate Geometry Objectives

ABC is a triangle. We will find its area. Construct points D and E so that ADEC is a trapezium. Coordinate Geometry Area of Triangles

ABC is a triangle. The vertices are arranged in an anticlockwise direction. We will find its area. Construct points D, E and F on the x-axis as shown. Coordinate Geometry

From the previous slide, we know that Definition Coordinate Geometry

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 5

Find the area of a quadrilateral with vertices A(x 1, y 1 ), B(x 2, y 2 ), C(x 3, y 3 ) and D(x 4, y 4 ), following an anticlockwise direction. Split the quadrilateral into two triangles. Coordinate Geometry Area of Quadrilaterals

The area of a quadrilateral with vertices A(x 1, y 1 ), B(x 2, y 2 ), C(x 3, y 3 ) and D(x 4, y 4 ), following an anticlockwise direction. The method for finding the area of quadrilaterals is very similar to that of triangles. Coordinate Geometry

Find the area of a quadrilateral with vertices P(1, 4 ), Q(–4, 3), R(1, –2) and S(4, 0), following an anticlockwise direction. Coordinate Geometry Exercise 7.2, qn 2(b)

7.3 Parallel and Non-Parallel Lines In this lesson, 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

The diagram shows a parallelogram ABCD with A and C on the x-axis and y-axis respectively. The equation of AB is x + y = 2 and the equation of BC is 2y = x (a) Find the coordinates of A, B and C. Coordinate Geometry Example 7(a)

The diagram shows a parallelogram ABCD with A and C on the x-axis and y-axis respectively. The equation of AB is x + y = 2 and the equation of BC is 2y = x (b) Find the equations of AD and CD. AD is parallel to BC (2y = x + 10). Gradient of AD = gradient of BC = 0.5 Since A is (2, 0) CD is parallel to AB (x + y = 2 ). Gradient of CD = gradient of AB = –1 Since C is (0, 5) Coordinate Geometry Example 7(b)

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 Exercise 7.3, qn 2(b)

Coordinate Geometry 7.4 Perpendicular Lines In this lesson, 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 14

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 Exercise 7.4, qn 9