Define what it means for two shapes to be mathematically similar .

Two shapes are mathematically similar if they have the same shape and their corresponding sides are in proportion. One is an enlargement of the other, so every length is multiplied by the same scale factor.

True or False? If the length scale factor squared equals the area scale factor, the two shapes must be similar.

False. The relationship is a test that can only ever disprove similarity, never prove it. Shapes that really are similar must satisfy it, but satisfying it is not enough on its own to guarantee that they are.

National 5MathsSQAFlashcardsGeometric SkillsSimilarity

Similarity (SQA National 5 Maths): Flashcards

Exam code: X847 75

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Define what it means for two shapes to be mathematically similar.

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Define what it means for two shapes to be mathematically similar.
Two shapes are mathematically similar if they have the same shape and their corresponding sides are in proportion.
One is an enlargement of the other, so every length is multiplied by the same scale factor.
A pair of similar shapes has length scale factor $k$ , so complete the other two scale factors.
$area scale factor =______$
$volume scale factor =______$
The completed scale factors are:
$area scale factor = k^{2}$
$volume scale factor = k^{3}$
The powers match the number of dimensions, so a length has one, an area two and a volume three.
How do you work out a scale factor between two similar shapes?
Divide a quantity on one shape by the corresponding quantity on the other, keeping the same order throughout.
Heights of $15$ cm and $10$ cm give a length scale factor of $\frac{15}{10} = \frac{3}{2}$ .
Solid $B$ is $1.5$ times as tall as solid $A$ so how much greater is its volume?
Cube the length scale factor, so ${(\frac{3}{2})}^{3} = \frac{27}{8}$ .
A volume of $32 {cm}^{3}$ therefore becomes $32 \times \frac{27}{8} = 108 {cm}^{3}$ .
How do you get the length scale factor from an area or a volume scale factor?
Take the square root of an area scale factor, or the cube root of a volume scale factor.
Once you have $k$ you can raise it to whichever power the quantity you actually want needs.
What does a scale factor between $0$ and $1$ tell you?
The shape is getting smaller, since multiplying by a number less than one reduces every length.
The same holds for the area and volume scale factors, because $k^{2}$ and $k^{3}$ then also lie between $0$ and $1$ .
True or False?
If the length scale factor squared equals the area scale factor, the two shapes must be similar.
False.
The relationship is a test that can only ever disprove similarity, never prove it.
Shapes that really are similar must satisfy it, but satisfying it is not enough on its own to guarantee that they are.
How do you show that two cups are not mathematically similar?
Work out the length scale factor, cube it, and compare that with the actual volume scale factor.
Heights of $13.5$ cm and $12$ cm give ${(\frac{9}{8})}^{3} = 1.42$ , while volumes of $440$ and $330$ give $\frac{4}{3} = 1.33$ , so the two do not match.

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