Ratio and Proportion in Similarity for the ESAT

The Definition of Similarity

In mathematics, two shapes are considered similar if one is a perfect enlargement of the other. For similarity to exist, the shapes must be the same shape and have the same interior angles in the same order. Consequently, the ratios of all corresponding sides must be equal. If two shapes are similar, they are not necessarily the same size, but their proportions are identical.

Area and Volume Ratios from a Linear Scale Factor

When we compare two similar shapes, A and B, we often use a linear scale factor, which we will call $x$. This factor describes how many times larger the lengths of shape B are compared to the lengths of shape A. The relationships for higher dimensions are as follows:

1. The lengths of the sides of B are $x$ times the corresponding lengths of A.
2. The area of the surfaces of B are $x^2$ times the corresponding areas of A.
3. The volume of B is $x^3$ times the volume of A.

Ratios in Similarity

We can also express these relationships using ratio notation. If the ratio of corresponding lengths on shape A to those on shape B is $x : y$, then:

1. The ratio of corresponding areas is $x^2 : y^2$.
2. The ratio of the volumes is $x^3 : y^3$.

Finding Linear Ratios from Area or Volume

To find the linear relationship when only the area or volume is known, we must reverse the process using roots.

Linear Ratio from the Area Ratio

If the area of shape B is $x$ times the area of shape A, then the lengths of shape B are $\sqrt{x}$ times the lengths of shape A. Similarly, if the ratio of the area of B to the area of A is $x : y$, the ratio of the lengths is $\sqrt{x} : \sqrt{y}$.

Linear Ratio and Area Ratio from the Volume Ratio

If the volume of shape B is $x$ times the volume of shape A, then:

1. The lengths of shape B are $\sqrt[3]{x}$ times the corresponding lengths of shape A.
2. The areas on shape B are $(\sqrt[3]{x})^2$ times the corresponding areas on shape A.

If the ratio of the volume of B to the volume of A is $x : y$, then the ratio of lengths is $\sqrt[3]{x} : \sqrt[3]{y}$, and the ratio of areas is $(\sqrt[3]{x})^2 : (\sqrt[3]{y})^2$.

Worked Example: Calculating Volume from a Scale Factor

Consider two cubes, X and Y. Cube X has a side length of $x$ cm, and Cube Y has a side length of $3x$ cm. If the volume of X is $100$ cm$^3$, what is the volume of Y?

First, find the linear scale factor. We divide the length of Y by the length of X: $3x \div x = 3$. Since the linear scale factor is $3$, the volume of Y must be $3^3$ times the volume of X. Therefore, the volume of Y is $27 \times 100 = 2700$ cm$^3$.

Worked Example: Calculating Surface Area from a Linear Ratio

The ratio of the diameters of two spheres, A and B, is $2 : 5$. If the surface area of A is $250$ cm$^2$, find the surface area of B.

A diameter is a measure of length. Since the ratio of lengths is $2 : 5$, the ratio of areas is $2^2 : 5^2$, which simplifies to $4 : 25$ or $1 : 6.25$. To find the area of B, we multiply the area of A by $25/4$. The calculation is $25/4 \times 250 = 1562.5$ cm$^2$.

Worked Example: Calculating Length from an Area Ratio

Shapes A and B are mathematically similar. The ratio of the area of shape A to the area of shape B is $9 : 25$. If the length of shape A is $21$ cm, what is the length of shape B?

First, find the linear ratio by taking the square root of the area ratio. The linear ratio is $\sqrt{9} : \sqrt{25}$, which is $3 : 5$. Since the length of A is $21$ cm, we find the length of B by dividing $21$ by $3$ and multiplying by $5$: $(21 \div 3) \times 5 = 35$ cm.

Worked Example: Finding Area from a Volume Ratio

The volume of a cylinder, P, is $64$ times the volume of a mathematically similar cylinder, Q. The surface area of Q is $300$ cm$^2$. What is the surface area of P?

To find the area ratio, we must first determine the linear ratio. The ratio of volumes (Q to P) is $1 : 64$. Therefore, the ratio of corresponding lengths is $\sqrt[3]{1} : \sqrt[3]{64}$, which is $1 : 4$. Once we have the linear ratio of $1 : 4$, we can find the area ratio by squaring it: $1^2 : 4^2$, which is $1 : 16$. If the area of Q is $300$ cm$^2$, then the area of P is $300 \times 16 = 4800$ cm$^2$.

Similarity and Trigonometric Ratios

Similarity is closely linked to trigonometry. If triangle ABC is a right angled triangle with sides $x, y,$ and $z$, and triangle DEF is an enlargement of ABC with a scale factor $p$, the interior angles and the trigonometric ratios do not change. They are preserved because they depend only on the relative proportions of the sides, not their absolute lengths.

In the diagrams above, $\tan C = x/z$. For the enlarged triangle DEF, $\tan F = px/pz$. Since the factor $p$ cancels out, $\tan F = x/z$. This proves that $\tan F = \tan C$, meaning angle $C =$ angle $F$ as they are both acute. This same logic applies to sine and cosine ratios.

Worked Example: Sine Ratios in Enlarged Triangles

Triangle DEF is an enlargement of triangle ABC with a scale factor of $4$. The sides of triangle ABC have lengths $x, y,$ and $z$, where $y < x < z$. In triangle DEF, the side lengths satisfy $EF < ED < DF$. What is the sine of angle DFE in terms of $x, y,$ and $z$?

Because DEF is an enlargement of ABC, the triangles are similar and their corresponding angles are equal. Side $y$ is the smallest side in ABC, and $EF$ is the smallest side in DEF. Therefore, angle A (opposite $y$) corresponds to angle D (opposite $EF$). This means angle C must correspond to angle F. Since $\sin F = \sin C$, and in triangle ABC, $\sin C = x/z$, it follows that $\sin F = x/z$.