Similar Figures (VCE SSCE General Mathematics): Revision Notes

Similar Figures

What are similar figures?

Similar shapes share the same overall form but differ in size. Think of them as enlargements or reductions of each other - like looking at the same image through different magnifications.

These three frogs are similar figures. They have identical shapes but are different sizes.

Conditions for similarity

For two polygons (closed shapes with straight sides) to be similar, they must satisfy two essential conditions:

Let's examine these two rectangles. They are similar because:

• All their corresponding angles are equal (all are $90°$ in both rectangles)
• Their corresponding sides are in the same ratio

For the side lengths:

• Ratio of longer sides = $6 : 3$ which simplifies to $2 : 1$
• Ratio of shorter sides = $4 : 2$ which simplifies to $2 : 1$

Both ratios simplify to the same value ($2:1$), confirming the sides are proportional.

Scale factor for lengths

When shapes are enlarged or reduced, we use a scale factor to describe the relationship between them. This tells us how much bigger or smaller the new shape is compared to the original.

Calculating scale factor

The scale factor is calculated using this formula:

For the rectangles shown earlier:

$k = \frac{6}{3} = 2$

This means Rectangle A has been scaled up by a factor of $2$ to create Rectangle B.

Scaling up and scaling down

Worked example: Finding scale factor of length

Scale factor for areas

When you scale a shape's lengths by a factor of $k$, something interesting happens to its area - it doesn't scale by $k$, but by $k^2$ (k squared).

The relationship between length and area scale factors

Let's see why this works using our rectangles from earlier:

• Area of small rectangle (Rectangle A) = $3 \text{ cm} \times 2 \text{ cm} = 6 \text{ cm}^2$
• Area of large rectangle (Rectangle B) = $6 \text{ cm} \times 4 \text{ cm} = 24 \text{ cm}^2$

The length scale factor was $k = 2$.

For the area scale factor:

$k^2 = 2^2 = 4$

We can verify this:

$\frac{\text{Area of Rectangle B}}{\text{Area of Rectangle A}} = \frac{24}{6} = 4$

So when the lengths doubled ($k = 2$), the area became four times larger ($k^2 = 4$).

Another example with triangles

Consider two similar triangles where the length scale factor is $3$:

• Small triangle area = $6 \text{ cm}^2$
• Large triangle area = $54 \text{ cm}^2$

The area scale factor should be $k^2 = 3^2 = 9$.

Let's verify:

$\frac{54}{6} = 9$

The area has indeed been multiplied by $9$, confirming our rule.

Worked example: Finding scale factor of area

Continuing with the rectangles from the previous example:

Using similar triangles to find unknown values

One practical application of similar figures is finding unknown measurements. When two triangles are similar, we can use the proportional relationship between their sides to calculate missing lengths.

Setting up the proportion

Since corresponding sides are proportional in similar triangles, we can write:

$\frac{\text{side in triangle 1}}{\text{corresponding side in triangle 2}} = \frac{\text{another side in triangle 1}}{\text{corresponding side in triangle 2}}$

Worked example: Finding an unknown side length