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Coordinate Geometry

- The Cartesian Plane and Gradient

Basic Terminology

- The figure on the right shows 2 perpendicular

lines intersecting at the point O. This is called

the Cartesian Plane. - O is also called the origin.
- The horizontal line is called the x-axis and the

vertical line is called the y-axis

Coordinates of a Point

- The position of any point in the Cartesian Plane

can be determined by its distance from each axes.

- Example Point A is 3 units to the right of the

y-axis and 1 unit above the x-axis, its position

is described by the coordinate (3, 1). - Similarly, the coordinates of Points B, C and D

are determined as shown.

Question

- What coordinate represents the origin O ?

- Ans (0, 0)

Summary

- Any point, P, in the plane can be located by its

coordinate (x, y). - We call x the x - coordinate of P and y the

y - coordinate of P. - Hence, we say that P has coordinates (x, y).

Solution to Exercise 1

Gradient (or slope)

- The steepness of a line is called its GRADIENT

(or slope). - The gradient of a line is defined as the ratio of

its vertical distance to its horizontal distance.

Examples of Gradient

What is the gradient of the driveway?

Ans

Note Gradient has no units!

Examples of Gradient

An assembly line is pictured below. What is the

gradient of the sloping section?

Ans

Examples of Gradient

The bottom of the playground slide is 2.5 m from

the foot of the ladder. The gradient of the line

which represents the slide is 0.68. How tall is

the slide?

Ans

Question

- For safety considerations, wheelchair ramps are

constructed under regulated specifications. One

regulation requires that the maximum gradient of

a ramp exceeding 1200 mm in length is to be

- (a) Does a ramp 25 cm high with a horizontal

length of 210 cm meet the requirements? - (b) Does a ramp with gradient meet the

specifications? - (c) A 16 cm high ramp needs to be built. Find

the minimum horizontal length of the ramp

required to meet the specifications.

Ans No

Ans Yes

Ans 224 cm

Horizontal and Vertical Lines

- The gradient of a horizontal line is ZERO
- (Horizontal line is flat No Slope)
- The gradient of a vertical line is INIFINITY
- (Vertical line gradient is maximum)

Finding the gradient of a straight line in a

Cartesian Plane

- (a) Positive Gradients
- Lines that climb from left to the right are said

to have positive gradient/slope - (b) Negative Gradients
- Lines that descend from left to the right are

said to have negative gradient/slope

Examples

Write down the coordinates of the points given

(16, 0)

(0, 10)

(0, -8)

(-15, 0)

Examples

(-4, 0)

(0, 6)

(3, 0)

(0, -12)

Summary

Infinite

Gradient Formula

- So far, we have determined the gradient using the

- idea of
- Using the above, we must always remember to add a

negative sign to slopes with negative gradient. - Now, lets look at the formula to determine

gradient. The formula will take into

consideration the sign of the slope

Gradient Formula

How to apply gradient formula

- Write down the coordinates of 2 points on the

line (x1, y1) and (x2, y2) - If the coordinate is negative, include its sign
- Apply the formula

Examples

(1, 4)

L1 2 points on the line are (1, 4) and (0, 1)

(0, 1)

Tip Choose points that are easy to read!

1 square represents 1 unit on both axes

Examples

L2 2 points on the line are (1, 1) and (3, 3)

Tip Choose points that are easy to read!

1 square represents 1 unit on both axes

Examples

L3 2 points on the line are (3, 1) and (1, 0)

1 square represents 1 unit on both axes

Examples

L4 2 points on the line are (3, -1) and (-3,

-3)

1 square represents 1 unit on both axes

Examples

L5 2 points on the line are (0, 1) and (1, -2)

1 square represents 1 unit on both axes

Examples

L6 2 points on the line are (0, 0) and (-4, 4)

1 square represents 1 unit on both axes

Examples

L7 2 points on the line are (4, -2) and (-2, 2)

1 square represents 1 unit on both axes

Examples

L8 2 points on the line are (0, -2) and (-3,

-1)

1 square represents 1 unit on both axes

Question

- Is there a difference between

Ans No.

- Is there a difference between

Ans Yes!

Solution to Exercise 2

In order from smallest to largest gradient e, b,

a, d, c

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Horizontal Line Zero Vertical Line Inifinity