Similarity and Enlargement
Geometry · Similarity and Enlargement
Syllabus tag: KCSE | Mathematics | Form 2 | Topic 7 Similarity and Enlargement
Lesson objectives
By the end of this topic, you should be able to:
- State the conditions for two figures to be similar and construct similar figures.
- State the properties of enlargement as a transformation.
- Apply the properties of enlargement to construct objects and images.
- Relate the area and volume scale factors to the linear scale factor.
Similarity and Enlargement
Two figures are similar when one is an exact scaled copy of the other.
a) Conditions for similarity
Corresponding angles are equal.
Corresponding sides are in the same ratio.
For triangles, either condition alone is enough. If the angles match, the sides must be in proportion, and the other way round.
All circles are similar. All squares are similar. Rectangles are not automatically similar, since their side ratios can differ.
b) Congruent against similar
Congruent figures are identical in shape and size. The scale factor is 1.
Similar figures have the same shape but may differ in size.
So all congruent figures are similar, but not the other way round.
c) Enlargement
An enlargement is defined by a centre and a scale factor.
Each image point lies on the line from the centre through the object point.
Its distance from the centre is the scale factor times the object's distance.
d) Reading the scale factor
If k > 1, the image is larger. If 0 < k < 1, it is smaller, though the transformation is still called an enlargement.
If k is negative, the image is on the opposite side of the centre and is inverted.
If k = 1, the image and object coincide.
e) Finding the centre and factor
Given an object and its image, join corresponding points with straight lines. Those lines all meet at the centre.
The scale factor is any image length divided by the matching object length.
Check with a second pair of points. Both must give the same factor.
f) Area and volume scale factors
This is the part most often got wrong.
If lengths multiply by k, then areas multiply by k² and volumes multiply by k³.
For k = 3: a length triples, but the area becomes nine times and the volume twenty-seven times.
The reason is direct. Area multiplies two lengths, so it picks up two factors of k. Volume multiplies three.
g) Working backwards
Two similar solids have volumes in the ratio 8 : 27. The linear scale factor is the cube root, so 2 : 3.
If areas are in the ratio 16 : 25, take the square root. The linear factor is 4 : 5.
h) Where this is used
Scale models and maps. Photographic enlargement. Comparing the cost of tins of different sizes, where volume grows faster than surface area.
Words to know
- Similar figures -- figures with equal corresponding angles and sides in a constant ratio.
- Linear scale factor -- the ratio of an image length to the corresponding object length.
- Centre of enlargement -- the fixed point from which an enlargement is measured.
- Area scale factor -- the square of the linear scale factor.
- Volume scale factor -- the cube of the linear scale factor.
:::checkpoint Check yourself
- State the two conditions for two figures to be similar.
- Two similar triangles have sides in the ratio 2 : 5. What is the ratio of their areas?
- Two similar solids have volumes in the ratio 27 : 64. Find the linear scale factor.
- What happens to an image when the scale factor is negative? :::
Bridge to practice
The exercises begin with the conditions for similarity and the scale factor, move through enlargement and the meaning of negative and fractional factors, and finish with area, volume and applications. Whenever a question gives an area or volume ratio, take the square or cube root before applying anything to a length.