Term 3 : Unit 2 Coordinate Geometry Name : _____________ ( ) Class : ________ Date : ________ 2.1 Midpoint of the Line Joining Two Points 2.2 Areas of Triangles and Quadrilaterals 2.3 Parallel and Non-Parallel Lines 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 2.4 Perpendicular Lines 2.5 Circles 1

Coordinate Geometry Objectives 2.1 Midpoint of the Line Joining Two Points Objectives In this lesson, you will learn how to find the midpoint of a line segment and apply it to solve problems. 7.1 Midpoint of the Line Joining Two Points Objectives In this lesson you will learn how to find the midpoint of a line segment and apply it to solve problems.

Coordinate Geometry D is and E is M is the point A line AB joins points (x1, y1) and (x2, y2). M (x, y) is the midpoint of AB. Take the x-coordinate of D and the y-coordinate of E. Construct a right angled triangle ABC. Page 138 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

Coordinate Geometry Example 1 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. Example 4 Page 140

Coordinate Geometry Example 2 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. Exercise 7.1 Page 140 Question 3 Let M be the midpoint of DE.

Coordinate Geometry Example 3 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. Let M be the midpoint of AC. Exercise 7.1 Page 141 Question 4 M is also the midpoint of BD.

Coordinate Geometry Objectives 2.2 Areas of Triangles and Quadrilaterals Objectives In this lesson, you will learn how to find the areas of rectilinear figures given their vertices. 7.2 Areas of Triangles and Quadrilaterals Objectives In this lesson you will learn how to find the areas of rectilinear figures given their vertices.

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

Construct points D, E and F on the x-axis as shown. Coordinate Geometry 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. Example

From the previous slide, we know that Coordinate Geometry Definition Example From the previous slide, we know that

Coordinate Geometry Example 4 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. Example 5 Page 142

Split the quadrilateral into two triangles. Coordinate Geometry Area of Quadrilaterals Find the area of a quadrilateral with vertices A(x1, y1 ), B(x2, y2 ), C(x3, y3 ) and D(x4, y4 ), following an anticlockwise direction. Example Page 143 Split the quadrilateral into two triangles.

Coordinate Geometry The area of a quadrilateral with vertices A(x1, y1 ), B(x2, y2 ), C(x3, y3 ) and D(x4, y4 ), following an anticlockwise direction. Example Page 143 The method for finding the area of quadrilaterals is very similar to that of triangles.

Coordinate Geometry Example 5 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. Exercise 7.2 Question 2 (b) Page 144

Coordinate Geometry Objectives 3.3 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. 7.3 Parallel and Non-Parallel Lines Objectives In this lesson you will learn how to apply the conditions for the gradients of parallel lines to solve problems.

Coordinate Geometry Consider the straight line with equation y = m1 x + c1 that makes an angle of θ1 with the positive x-axis. Translate the line parallel to the x-axis. The new line has equation y = m2 x + c2 and makes an angle of θ2 with the positive x-axis. Page 142 The lines are parallel to each other. The lines make the same angle with the x-axis. θ1 = θ2 The lines have the same gradient. m1 = m2

Coordinate Geometry Example 6(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 + 10. (a) Find the coordinates of A, B and C. Example 7(a) Page 146

Coordinate Geometry Example 6(b) 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 + 10. (b) Find the equations of AD and CD. AD is parallel to BC (2y = x + 10). CD is parallel to AB (x + y = 2 ). Example 7(b) Page 146 Gradient of AD = gradient of BC = 0.5 Gradient of CD = gradient of AB = –1 Since A is (2, 0) Since C is (0, 5)

Coordinate Geometry Example 7 Find the equation of the line which passes through the point (–2, 3) and is parallel to the line 2x + 3y – 3 = 0. m in y = mx + c is the gradient of the line. Rearrange in the form y = mx + c. Exercise 7.3 Question 2 (b) Page 149

Coordinate Geometry Objectives 2.4 Perpendicular Lines In this lesson, you will learn how to apply the conditions for the gradients of perpendicular lines to solve problems. 7.4 Perpendicular Lines Objectives In this lesson you will learn how to apply the conditions for the gradients of perpendicular lines to solve problems.

Applies to any two perpendicular lines. Coordinate Geometry Consider the straight line with equation y = m1 x + c1 that makes an angle of θ1 with the positive x-axis. Rotate the line clockwise through 90°. The new line has equation y = m2 x + c2 and makes an angle of θ2 with the negative x-axis. Page 151 Applies to any two perpendicular lines.

Solve simultaneous equations Coordinate Geometry Example 8 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. Solve simultaneous equations Find P. Adding the equations Example 14 Page 155 Substitute for y

Coordinate Geometry Example 9 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. Find the equation of the perpendicular bisector of AB. (b) Find the coordinates of P and Q. Exercise 7.4 Question 9 Page 158

Curves and Circles Objectives 2.5 Circles In this lesson, you will learn to recognise the equation of a circle, find the centre and radius of a circle, find the intersection of a circle and a straight line. 9.2 Circles. Objectives In this lesson you will learn to recognise the equation of a circle, find the centre and radius of a circle and find the intersection of a circle and a straight line.

Curves and Circles Consider the point C(a, b) and a point P(x, y). The distance between C and P is Page 191 The locus of P as the line rotates around C is a circle of radius r. The equation of the circle is

Curves and Circles The circle with centre C(a, b) and radius r has equation Page 192 The general form of the equation is

The radius is the y-coordinate of C. Curves and Circles Example 10 Find the equation of the circle whose centre is C(–3, 4) and which touches the x–axis. The radius of the circle is 4 units. The equation of the circle is: Example 5, Page 193 The radius is the y-coordinate of C.

Substitute for x into the circle equation. Curves and Circles Example 11 Find the coordinates of the points of intersection of the line 2y + x = 12 with the circle x2 + y2 – 6x – 4y – 12 = 0. Substitute for x into the circle equation. Example 6, Page 193 Using x = 12 – 2y. The points are

Curves and Circles Example 12 (a) Give the equation of the circle with centre C(–2, 0) and radius r = 3. The equation is (b) Find the coordinates of the centre and the radius of the circle x2 + y2 – 2x – 6y + 1 = 0. Exercise 9.2, Question 1(a) and 2(a), Page 194