Stretches

If we enlarge (or reduce) an object only in one direction, it is called a stretch!

Mathematically speaking, there are two parameters we need to know:

• Stretch Factor: how much we stretch by. Usually given the value k
• Invariant Line: the line which we stretch away from. Think of it as the ‘unchanging line’.

Example:

If the invariant line is the y-axis, only the x-coordinate is affected by the stretch factor k.

(x,y) –> (kx,y)

And vice cersa for if the invariant line is the x-axis, only the y-coordinate changes.

(x,y) –> (x,ky)

Stretching with the x or y-axes as the IL is easy enough. However, if the IL is slightly off the x or y-axes, you have to count the distance of each point from the IL. Then you stretch according to that distance.

Look at the example below:

As you can see, the IL is the horizontal line with equation y=-2. Step-by-step, let’s look at point P first.

• P has the coordinate (4,0). Count: how far from the IL is P?
  
  Answer…P is 2 units away from the IL!
• What is the stretch factor? k=1/2
• That means I multiply the distance 2 by 1/2. Obviously, that will be 1.
• Therefore, P stretched with a factor of k=1/2 would have a new position of (4, -1).
  Why -1 ? Because the stretch factor is less than 1, so the stretch is going to be a stretching towards the IL. Like it’s squishing downwards!

P(4,0) –> P'(4,-1)

Let’s try point Q, then you should be able to do points R and S yourself.

As you can see, point Q is below the IL. Doesn’t matter. Just count the distance away again.

• The coordinate of Q is (8,-5).
• That means Q is 3 units away from the IL. The stretch factor is 1/2, so the new distance has to be 1.5 units away from the IL.
• Only the y-coordinate will be stretched by k=1/2. So the new position will be (8,-3.5)

Q(8,-5) –> Q'(8,-3.5)

Noice.

Try R and S yourself, and check the answer below!

Now, the real challenge is in trying to solve these kinds of problems without drawings. You can try to visualise it, or draw it based on the info given. For example:

Answers…coming soon!

Enlargements

Sometimes, we like to make an object smaller or larger than its original size.

This is similar to rotation, in that we need to know two pieces of information:

• Centre of Enlargement (a point somewhere away from the image, usually the origin)
• Scale factor (usually called k)

The best way to do perform an enlargement is to count how far away each point on the object is from the centre of enlargement, and then multiply by the scale factor.

For example, look at the triangle ABC below.

We are going to enlarge from centre (0,0). As you can see, A is at the point (1,1). If we use scale factor 2, then A’ will be twice the distance of A from O. Luckily this is just (1,1) so you can easily located A’ at (2,2)! Have a look at points B and C and make sure you understand how that works.

HINT: If the centre of enlargement is the origin, it’s quite easy, just multiply the coordinates by the scale factor. But it won’t always be from the origin!

Note, that if we have a scale factor of less than 1 (i.e. a fraction), then the image will be smaller – in other words, a reduction.

Textbook Ex: (you should try and copy these onto paper for better results!)

ANSWERS BELOW

3a) (4.5, 6)
3b) (0,2)

Rotations

Rotations is exactly what it sounds like. Take an object and then spin it around. But what’s important is where do you spin around?

Consider a merry-go-round. All the horses are rotated around the centre obviously right? In transformations, we give this point a special name: the centre of rotation.

So take the object, identify the centre of rotation, and spin around! In reality, using tracing paper is the most effective way to do it when asked to draw rotations.

The two parameters we need are:

• Centre of rotation – (x and y coordinate)
• Angle of rotation – usually given as θ

Step by step:
1) Identify your object
2) Place corner of tracing paper on the centre of rotation
3) Mark your object point on the tracing paper
4) Rotate the tracing paper θ degrees in the correct direction
5) Mark your image point on the number plane

Note: (Tracing paper is provided during exams. If you don’t have tracing paper, you can use normal paper, but obviously it’s a little more challenging to see)

Once you have enough practice, you may be able to visualise it mentally and plot the point accordingly.

Try this Geogebra tool to rotate the object around 90°, 180° and 270°. Create your own object and try changing the centre of rotation to see how it affects your image.

Myimaths Exercises
– Lesson
– Online Homework

If you have access to a printer, try these worksheets.

Click here to return to the Transformations home page.

Year 11: Transformations Unit

In this unit, we will be looking at mathematical techniques of changing shapes and figures. They are as follows:

• Translations
• Rotations
• Reflections
• Enlargements
• Stretches

The first 3 are relatively easy, with enlargements and stretches being slightly more challenging. 

You will need: 

• Drawing skills
• Drawing tools: pencil, ruler paper, tracing paper
• Your brain

Parallel Lines and Converse Proofs (year 9 Continued)

In the same way that we use parallel lines to find certain angles, we can also use certain angles to prove two lines are parallel. Here is the worksheet we did together today. Can you help this student fix their mistakes? 🙂

NOTE: Be careful with your spelling! It’s also acceptable to use abbreviations/shorthand, as long as it is clear which property you are using.

E.g.

• Corresponding angles = corr. ∠s
• Alternate angles = alt. ∠s
• Co-interior angles = coint. ∠s
• Vertically opposite angles = vert. oppos. ∠s
• Angles on a straight line = ∠s on a straight line, linear pair, adj. ∠s