Video: AQA GCSE Mathematics Higher Tier Pack 5 • Paper 1 • Question 23

A rectangle is drawn on the grid. a) The rectangle is reflected in the line 𝑦 = 3. Circle the number of invariant vertices of the rectangle. [A] 0 [B] 1 [C] 2 [D] 4 b) The rectangle is enlarged by a scale factor of 2 with a centre of (2, 2). Circle the number of invariant vertices of the rectangle. [A] 0 [B] 1 [C] 2 [D] 4 c) The rectangle is translated by the vector (1 and 2). Circle the number of invariant vertices of the rectangle. [A] 0 [B] 1 [C] 2 [D] 4

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Video Transcript

A rectangle is drawn on the grid. a) The rectangle is reflected in the line 𝑦 equals three. Circle the number of invariant vertices of the rectangle. b) The rectangle is enlarged by a scale factor of two with a centre of two, two. Circle the number of invariant vertices of the rectangle. c) The rectangle is translated by the vector one, two. Circle the number of invariant vertices of the rectangle.

Let’s begin by recalling what we mean by the word invariant. Invariant means unchanged. For each question, we’ll need to consider the number of vertices — remember those are just corners of the shape — that remain unchanged after the transformation. For part a, we’re going to reflect the triangle in the line 𝑦 equals three. This is the line for which all of its coordinates have 𝑦-values of three. It’s a horizontal line that passes through the 𝑦-axis at three.

To reflect the rectangle in this line, we’ll reflect each individual vertex. Notice that two vertices — that’s the one at two, three and six, three — lie on this line. They’ll actually remain where they are. The vertex at two, two though is one square away from the mirror line. Its image will also be one square away on the other side of the line; it’s here. Similarly, the vertex at six, two is one square away from the mirror line. Its image will end up one square away from the mirror line on the other side; it’s here. The image of the reflected rectangle is as shown. We can see that there are exactly two invariant vertices of this rectangle. They lie at two, three and at six, three.

Now let’s consider part b. To enlarge this shape by scale factor of two about the centre two, two, we’ll work out how far away each vertex is from the centre and then multiply that by two. Notice that one of the vertices lies on the centre itself. Since the distance from the centre here is zero, when we double this distance it will still be zero. So this point remains unchanged. This vertex is invariant. We should be able to spot that only one of the vertices will be invariant. But let’s see where the rest end up.

The point two, three is one square away from the centre. When we enlarge this shape, we need to double it. And in the same direction, we need to be two squares away from the centre; that’s here. The point six, two is four squares away from the centre. When we double this, we need to be eight squares away from the centre in the same direction. That’s here: 10, two. We can now spot that the enlarged shape is as shown. And again, we can see that the number of invariant vertices is one.

For part c, we’re going to translate the rectangle by the vector one, two. We need to move it one space right and two spaces up. The entire shape slides in that direction. This means that none of its vertices can be invariant. It’s zero.