  # Solve Geometrical Problems

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

Midpoints

Example

Find the mid lengths of AB and BD Mid point of line AB

Take both x co-ordinates and add 8 + 2 = 10. Now divide by 2 = 5

Take both y co-ordinates and add 6 + 6 = 12. Now divide by 2 = 6

Mid point = (5,6)

Mid point of line BD

Take both x co-ordinates and add 8 + 8 = 16. Now divide by 2 = 8

Take both y co-ordinates and add 6 + 2 = 8. Now divide by 2 = 4

Mid point = (8,4)

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

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

2 + 6 = 8 ÷ 2 = 4

7 + 5 = 12 ÷ 2 = 6

Mid point is (4,6)

Line Lengths  1. Write out the co-ordinates P = (5,3) and Q = (1, -2).

On the diagram you can see that these co-ordinates make a right angled triangle.

2.  Find the difference between the x co-ordinates.

3. The difference between x is 4 and the difference between y is 5.

4. Now apply Pythagoras's Theorem 4² + 5² = √ 41. 6.4 units correct to 1 decimal place.

Gradients are written in the form y = mx + c.  The gradient is represented by the letter m. Draw a right angle anywhere on the line. Count the units.

Gradient is rise ÷ run (vertical ÷ horizontal)

4 ÷ 5 = 1.5 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 It just means take the second y co-ordinate and subtract the first y co-ordinate 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. Let's get coordinated to find that treasure

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