18.2 Enlargement — Learn

Prerequisites: coordinates, multiplication and ratio.

Objectives: enlarge points and shapes by positive scale factors; find missing lengths using the scale factor.

An enlargement changes a shape's size from a fixed centre. The scale factor tells you how each length changes.

For an enlargement centred at the origin with positive scale factor $k$, multiply both coordinates by $k$.

Example. Enlarge $(2,3)$ by scale factor $2$ about the origin.

Multiply the $x$-coordinate by $2$: $2\times2=4$. Multiply the $y$-coordinate by $2$: $3\times2=6$. The image is $(4,6)$.

To find a scale factor from two corresponding lengths, divide image length by original length.

Example. A side of $4$ cm becomes $10$ cm.

Scale factor $=10\div4$. $10\div4=2.5$. The scale factor is $2.5$.

Guided investigation: coordinate changes

Predict: for a point $(1,1)$, predict its image after an enlargement by scale factor $2$ about the origin.

Test: move the scale-factor control and check the image. Explain: how does each coordinate change, and how can you check the distance from the origin?

Watch out. A scale factor of $2$ doubles every length, not every coordinate difference from an arbitrary point. Check that the stated centre is used.

Progress check

One mural tile side grows from $6$ cm to $9$ cm. Work out the scale factor. Then work out the matching length for a side that was $4$ cm on the smaller tile, and explain why the same multiplier applies.

You should now be able to

• enlarge points about the origin using a positive scale factor;
• identify how a stated centre affects the transformation;
• calculate a scale factor from corresponding lengths;
• use a scale factor to find missing lengths.