The centre of an enlargement is where the lines through matching object and image points meet. Draw two such lines to find it, and confirm with a third or with the scale factor.
This lesson is part of the section on congruency, enlargement and combined transformations. You need the ratio from calculating enlargement scale factors.
What are the steps?
Follow these steps on a grid or with coordinates.
- Match each object point to its image point, such as P to P’.
- Draw a straight line through each pair, and extend it.
- Mark where the lines cross. That point is the centre.
- Check that each image point is k times as far from the centre as its object point.
Worked example: a triangle on a grid
Object triangle PQR has P(2, 2), Q(3, 2) and R(2, 3). The image triangle has P’(4, 4), Q’(7, 4) and R’(4, 7).
Line through P and P’. It passes through (2, 2) and (4, 4), so its equation is y = x.
Line through R and R’. It passes through (2, 3) and (4, 7), which has gradient 2. Its equation is y − 3 = 2(x − 2), so y = 2x − 1.
Intersection. Set x = 2x − 1, so x = 1 and y = 1. The centre is (1, 1).
Check with Q and Q’. The line through (3, 2) and (7, 4) has gradient 1/2. At x = 1, y = 2 + (1 − 3)/2 = 1, so the line passes through (1, 1). All three lines meet at the same point.
The scale factor is k = P’Q’ ÷ PQ = 3 ÷ 1 = 3. The distance from (1, 1) to P(2, 2) is √2, and to P’(4, 4) is 3√2, which is 3 times as far.
The mistake that uses a single line
A common slip is to draw only one line, such as PP’, and place the centre somewhere on it. One line cannot fix the centre.
A second line gives the crossing point. If the third line misses it, recheck the matching of points.
Check yourself
The object points are A(2, 1) and B(1, 3). The image points are A’(4, 2) and B’(2, 6). Find the centre and the scale factor.
Answer
Line AA’ passes through (2, 1) and (4, 2), so its equation is y = x/2.
Line BB’ passes through (1, 3) and (2, 6), so its equation is y = 3x.
Set x/2 = 3x, which gives x = 0 and y = 0. The centre is (0, 0).
k = 4 ÷ 2 = 2. Check: B’ is (2, 6), which is twice (1, 3).
What to study next
Next, see how an enlargement combines with other transformations in describing the order of combined transformations. Plot your own points with the transformation coordinates explorer.
If you want a teacher to watch you locate centres on new diagrams, see online one-to-one Mathematics tuition.