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Solve Geometrical Problems Using Coordinates

In this worksheet, students will use coordinates to find midpoints, line lengths and gradients of lines.

'Solve Geometrical Problems Using Coordinates' worksheet

Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   OCR, AQA, Eduqas, Pearson Edexcel,

Curriculum topic:   Basic Geometry, Geometry and Measures

Curriculum subtopic:   Conventions, Notations and Terms Properties and Constructions

Difficulty level:  

Worksheet Overview

A battleship                                                       Battleships grid

 

Do you ever play Battleships? 

Did you ever find the treasure on that pirate map?

All great fun until you are asked to calculate midpoints of lines, their length or gradients.

 

So from Battleships to G.C.S.E questions, this is what we do:

 

Finding midpoints of lines

Example:

Find the midpoints of AB and BD

 

Find the midpoints of lines using coordinates

 

Midpoint of line AB:

Take both x coordinates and add 8 + 2 = 10. Now divide by 2 = 5

Take both y coordinates and add 6 + 6 = 12. Now divide by 2 = 6

Midpoint = (5,6)

 

Mid point of line BD:

Take both x coordinates and add 8 + 8 = 16. Now divide by 2 = 8

Take both y coordinates and add 6 + 2 = 8. Now divide by 2 = 4

Midpoint = (8,4)

 

You don't actually need a diagram, you can do this without:

Find the midpoint of AB.  A = (2,7)   B =  (6,5)

2 + 6 = 8 ÷ 2 = 4

7 + 5 = 12 ÷ 2 = 6

Midpoint is (4,6)

 

 

 Line Lengths

 

line on a coordinate grid   line length

1. Write out the coordinates:  P = (5,3) and Q = (1,-2)

 On the diagram, you can see that these coordinates make a right-angled triangle.

 

2.  Find the difference between the x coordinates.

The difference between x is 4  (5 - 1 = 4) and the difference between y is 5 (3 -- 2 is the same as 3 + 2 = 5)

 

3. Now apply Pythagoras's Theorem:   a² + b²  =  c²                      

4² + 5² =  √ 41

This is 6.4 units correct to 1 decimal place.

 

 

Gradients

Gradients are written in the form y = mx + c.  The gradient is represented by the letter m.

 

The gradient of a line

 

Draw a right angle anywhere on the line. Count the units.  

The gradient is rise ÷ run (vertical ÷ horizontal)

The vertical units = 5 and the horizontal units = 4

5 ÷ 4 = 1.25 units

 

Like the others, you can find a gradient without any diagram.

Find the gradient of the line with coordinate points A = (4,9) B = (6,13)

Here is the formula:

formula for gradients of a line

It just means take the second y coordinate and subtract the first y coordinate from it.

Do the same for the x coordinates and then divide.

13 - 9 = 4

6 - 4 = 2

4 ÷ 2 = 2

The gradient of the line is 2.

 

treasure chest

 

 Let's get coordinated to find that treasure.

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