Year 9AC9M9M03

Similar Triangles and Scale Factor

Solve spatial problems, applying angle properties, scale, similarity, Pythagoras' theorem and trigonometry in right-angled triangles.

Similar shapes are the same shape at a different size. Angles stay equal, sides all change by the same factor.

Method

Similar triangles

\text{AAA}

Three equal angles means similar. Two is enough, since the third follows.

Scale factor

k = \frac{\text{new side}}{\text{old side}}

The same $k$ for every pair of matching sides.

Matching sides

\frac{a}{a'} = \frac{b}{b'} = \frac{c}{c'}

Corresponding sides are in the same ratio.

Area scales by $k^{2}$

\text{new area} = k^{2} \times \text{old area}

Double the sides, four times the area.

e.g.

\text{a triangle } 3, 4, 5 \text{ is enlarged so the shortest side becomes } 9

Match the sides. The shortest goes with the shortest $3 \rightarrow 9$
Find the scale factor. Divide new by old $k = \tfrac{9}{3} = 3$
Multiply every other side by $k$ $4 \times 3 = 12 \quad 5 \times 3 = 15$
Check the angles are unchanged. They must be $36.87^\circ,\ 90^\circ,\ 53.13^\circ$
If asked for area, remember it scales by $k^{2}$ $9 \times \text{old area}$

Wrong

\text{Adding the same amount to every side}

Right

\text{Multiply every side by } k

Enlarging is multiplying, not adding. Adding 6 to each of 3, 4, 5 gives 9, 10, 11 — a different shape.

Wrong

\text{Area scales by } k

Right

\text{Area scales by } k^{2}

Double the sides and the area goes up four times, not two. Length is one direction, area is two.

Wrong

\text{Matching the wrong sides}

Right

\text{Shortest with shortest, longest with longest}

Order the sides first. Pairing them wrongly gives a different scale factor for each pair, which is the giveaway that something is off.

Find the scale factor and the missing side:

3,4,5 \rightarrow 6,?,?

Answer

k = 2:\ 8 \text{ and } 10

5,12,13 \rightarrow 15,?,?

Answer

k = 3:\ 36 \text{ and } 39

2,3,4 \rightarrow ?,9,?

Answer

k = 3:\ 6 \text{ and } 12

6,8,10 \rightarrow 3,?,?

Answer

k = \tfrac{1}{2}:\ 4 \text{ and } 5

7,24,25 \rightarrow 14,?,?

Answer

k = 2:\ 48 \text{ and } 50

\text{Sides double. What happens to the area?}

Answer

\text{It becomes } 4 \text{ times bigger}

\text{Sides treble. What happens to the area?}

Answer

\text{It becomes } 9 \text{ times bigger}

\text{A triangle of area } 5 \text{ is enlarged by } k = 4. \text{ New area?}

Answer

80