Fully describe the single transformation that maps trapezium 𝐴 to trapezium 𝐵.
There are four types of transformation that we need to be aware of. These are translation, reflection, rotation, and enlargement. A translation is where an object is just slid or moved to a different position on the coordinate grid. In a reflection, an object is flipped in a mirror line. A rotation turns an object about a centre of rotation. And an enlargement changes the size of an object. It’s made either larger or smaller.
Looking at the diagram, we can see straight away that trapeziums 𝐴 and 𝐵 are different sizes. And the only one of our four transformations which changes the size of an object is an enlargement. There are two pieces of information that we need to give when describing an enlargement. The first of these is the scale factor: how many times bigger or smaller the object has been made.
To do this, we can look at pairs of corresponding lengths on the two shapes. On the shape 𝐴, the base of the trapezium is two units. That’s the difference between the 𝑥-values of three and one. On shape 𝐵, it’s six units. That’s the difference between the 𝑥-values of nine and three.
To work out the scale factor formulae, we can divide the length on the enlarged shape or image by the corresponding length on the original shape, the object. So we divide six by two. And we found that the scale factor is three, meaning that all the lengths of shape 𝐵 are three times larger than the corresponding lengths on shape 𝐴.
We could check this by looking at other pairs of corresponding lengths on the two shapes, for example, the other parallel side, which have lengths of one and three units. So now we know that the transformation is an enlargement with a scale factor of three.
The final thing we need to work out is the centre of enlargement. This is the point from which the enlargement has occurred. To find this point, we draw on lines, also called rays, connecting corresponding vertices of the two shapes together. So here I’ve drawn on the first one, and here is the second.
We don’t actually need to draw on anymore. But it’s always good to have at least three rays just to be sure. So here’s the third. What you’ll notice is that these rays cross at a common point. And this is the centre of enlargement. This point is on the 𝑦-axis with a 𝑦-coordinate of one. So it has the coordinates zero, one.
We found then that the single transformation that maps trapezium 𝐴 to trapezium 𝐵 is an enlargement with a scale factor of three and a centre of zero one. Notice that word in the question “single.” It’s really important that we write down only one transformation that will map from 𝐴 to 𝐵 in a single step. If we were to write down a combination of transformations, then this would not be the correct answer.