Similarity

Core Concepts

Similar Figures Two geometric figures are said to be similar if they have the exact same shape, but not necessarily the same size. For two polygons to be similar, two conditions must be met:

1. All corresponding angles are equal.
2. The ratio of the lengths of their corresponding sides is constant. This constant ratio is known as the scale factor.

If polygon $A$ is similar to polygon $B$, we write $A \sim B$.

For triangles, checking similarity is simpler because we have specific similarity theorems.

Similarity Theorems for Triangles Triangles are the most commonly tested similar figures. To prove that two triangles are similar, you only need to show that one of the following theorems holds true:

1. AA (Angle-Angle) Similarity: If two angles of one triangle are equal to two angles of another triangle, then the two triangles are similar. (Because the sum of angles in a triangle is always $180^\circ$, the third angles will automatically be equal).
2. SSS (Side-Side-Side) Similarity: If the lengths of the corresponding sides of two triangles are proportional, then the triangles are similar. Mathematically, for triangles $\triangle ABC$ and $\triangle XYZ$, if $\frac{AB}{XY} = \frac{BC}{YZ} = \frac{CA}{ZX}$, then $\triangle ABC \sim \triangle XYZ$.
3. SAS (Side-Angle-Side) Similarity: If two sides of one triangle are proportional to two sides of another triangle, and their included angles are equal, then the triangles are similar.

Properties of Similar Triangles Once two triangles are proven to be similar with a scale factor of $k$, the following relationships hold:

• Ratio of corresponding sides $= k$
• Ratio of their perimeters $= k$
• Ratio of their areas $= k^2$
• Ratio of their volumes (for 3D similar solids) $= k^3$

Worked Examples

Example 1: Finding Missing Sides using the Scale Factor

Question: The sides of a triangle are $5\text{ cm}$, $7\text{ cm}$, and $9\text{ cm}$. If the longest side of a similar triangle is $27\text{ cm}$, find the lengths of the other sides.

Solution: Let the sides of the first triangle be $a = 5\text{ cm}$, $b = 7\text{ cm}$, and $c = 9\text{ cm}$. The longest side of the first triangle is $c = 9\text{ cm}$.

Let the sides of the second (similar) triangle be $a'$, $b'$, and $c'$. We are given the longest side of the second triangle, $c' = 27\text{ cm}$.

Because the triangles are similar, the ratio of their corresponding sides is constant. We first find the scale factor $k$: $$k = \frac{\text{Longest side of second triangle}}{\text{Longest side of first triangle}} = \frac{c'}{c}$$ $$k = \frac{27}{9} = 3$$

Now, apply the scale factor to find the other two sides: $$a' = k \times a = 3 \times 5\text{ cm} = 15\text{ cm}$$ $$b' = k \times b = 3 \times 7\text{ cm} = 21\text{ cm}$$

Answer: The lengths of the other sides of the similar triangle are $15\text{ cm}$ and $21\text{ cm}$.

Example 2: Proving Similarity and Finding Unknown Lengths

Question: In two right-angled triangles $\triangle PQR$ and $\triangle STR$ that meet at vertex $R$, the angle at $Q$ is $90^\circ$ and the angle at $T$ is $90^\circ$. Points $Q$, $R$, and $T$ lie on a straight horizontal line. The given lengths are $QR = 15\text{ m}$, $PQ = 20\text{ m}$, and $RT = 24\text{ m}$. Prove that $\triangle PQR \sim \triangle STR$ and find the length of $\overline{ST}$.

Solution: Step 1: Prove Similarity To prove that $\triangle PQR$ and $\triangle STR$ are similar, we use the AA (Angle-Angle) similarity theorem:

1. $\angle PQR = \angle STR = 90^\circ$ (Both are given as right angles).
2. $\angle PRQ = \angle SRT$ (Vertically opposite angles are equal, since the lines intersect at $R$).

Since two corresponding angles are equal, the third angles must also be equal ($\angle QPR = \angle TSR$). Therefore, by the AA Similarity theorem, $\triangle PQR \sim \triangle STR$. Note that the correct vertex matching is $P \leftrightarrow S$, $Q \leftrightarrow T$, and $R \leftrightarrow R$.

Step 2: Find the length of $\overline{ST}$ Because $\triangle PQR \sim \triangle STR$, the ratio of their corresponding sides is equal: $$\frac{QR}{TR} = \frac{PQ}{ST}$$

Substitute the known values into the ratio: $$\frac{15}{24} = \frac{20}{ST}$$

Cross-multiply to solve for $ST$: $$15 \times ST = 20 \times 24$$ $$15 \times ST = 480$$ $$ST = \frac{480}{15} = 32\text{ m}$$

Answer: The length of $\overline{ST}$ is $32\text{ m}$.

NECTA Exam Focus

General Trends and Testing Patterns In the NECTA CSEE, the topic of Similarity is frequently tested in Paper 1, especially in the short-answer geometric sections. The questions usually assess your ability to:

1. Apply Scale Factors: You are often given the dimensions of one triangle and a single dimension of a similar triangle. You must match the corresponding sides (e.g., shortest to shortest, longest to longest) to find the scale factor and solve for the unknown sides.
2. Prove Similarity: Students are expected to formally demonstrate that two geometric figures are similar. The Angle-Angle (AA) similarity theorem for triangles is the most heavily tested. Look out for intersecting lines that create vertically opposite angles, parallel lines that create alternate interior angles, or a smaller triangle nested inside a larger one sharing a common vertex.
3. Solve for Unknowns in Intersecting Triangles: Diagrams featuring two right-angled triangles sharing a vertex (like in the 2024 paper) appear frequently.

Common Pitfalls and Mistakes

• Incorrect Matching of Corresponding Sides: This is the most common error. When triangles are flipped or rotated, do not just match sides by how they "look". Instead, use the angles: the sides opposite the equal angles correspond to each other.
• Confusing Proving Similarity with Congruence: Remember that similarity implies sides are proportional, whereas congruence implies sides are exactly equal. Do not accidentally use congruence theorems (like proving equal side lengths) when you only need to prove equal angles.
• Skipping the Formal Proof: In questions asking you to "prove" similarity before finding a length, you must clearly write out the reasoning (e.g., stating which angles are equal and why) rather than jumping straight into the ratio calculations.

Practice Problems

1. [2022 Paper 1] The sides of a triangle are $4\text{ cm}$, $5\text{ cm}$ and $6\text{ cm}$. If the longest side of a similar triangle is $18\text{ cm}$, find the lengths of the other sides.
2. [2024 Paper 1] A firm reserved two triangular plots for the construction of more offices as shown in the following diagram: Two triangles $AEB$ and $CDE$ share vertex $E$. Triangle $AEB$ has $AE = 24\text{ m}$ (horizontal), $AB = 36\text{ m}$ (vertical), with a right angle at $A$. Triangle $CDE$ has $ED = 32\text{ m}$ (horizontal), with a right angle at $D$.
• Prove that the triangles $AEB$ and $CDE$ are similar.
• Using similarity properties, find the length of $\overline{CD}$ in metres.

Subtopics

• Similar figures
• Similarity theorems and applications

Crosswalk Notes

Cross-version relationships are drafted in data/curricula/crosswalks/csee-basic-mathematics-2005-to-mathematics-2023.json. Partial and 2005-only mappings remain reviewable.