Two figures are similar if:
1. All corresponding angles are equal, and
2. All corresponding sides are in the same ratio (proportion).
The ratio is called the scale factor $k$:
$$k = \frac{\text{image (new) length}}{\text{original length}}$$
| Measurement | Effect of scale factor $k$ |
|---|---|
| Length | Multiply by $k$ |
| Area | Multiply by $k^2$ |
| Volume | Multiply by $k^3$ |
Two similar triangles: Triangle 1 has sides 6 cm, 8 cm, 10 cm. Triangle 2 has a corresponding side of 15 cm (to the 6 cm side).
Scale factor: $k = \dfrac{15}{6} = 2.5$
Other sides of Triangle 2: \$8 \times 2.5 = 20$ cm and \$10 \times 2.5 = 25$ cm.
Two similar rectangles. The smaller has area \$12 \text{ cm}^2$. The scale factor from small to large is $k = 3$.
$$\text{Larger area} = 12 \times 3^2 = 12 \times 9 = 108 \text{ cm}^2$$
A scale diagram represents a real object or region at a reduced (or enlarged) size. The scale is expressed as a ratio, e.g., \$1 : 500$ meaning 1 cm on the diagram = 500 cm (5 m) in reality.
$$\text{Real distance} = \text{scale distance} \times \text{scale factor}$$
On a scale diagram with scale \$1:250$, a wall measures 4.8 cm.
$$\text{Real length} = 4.8 \times 250 = 1200 \text{ cm} = 12 \text{ m}$$
To confirm two triangles are similar, show one of:
- AA: Two pairs of equal angles (angle-angle)
- SSS: All three pairs of sides in proportion
- SAS: Two pairs of proportional sides and the included angle equal
APPLICATION: Scale diagrams are used in architecture, engineering, cartography, and surveying. VCAA questions may ask you to find a real measurement from a scale diagram or vice versa.
COMMON MISTAKE: Forgetting to square (or cube) the scale factor for area (or volume). Length $\times k$, area $\times k^2$, volume $\times k^3$.
