Similarity of Triangles: Concept & Conditions

Similarity of triangles is a fundamental concept in geometry that deals with the relationships between triangles that have the same shape, but not necessarily the same size. Two triangles are considered similar if their corresponding angles are equal and the ratio of their corresponding sides is constant. Understanding similarity is crucial for solving various geometric problems, including those involving proportions, scale drawings, and indirect measurement.

The key takeaway is that similar triangles have the same shape, meaning they look the same, but one might be an enlarged or reduced version of the other.

Formulae

There are several ways to determine if two triangles are similar. These conditions lead to specific ratios and relationships between sides and angles:

• Side-Side-Side (SSS) Similarity: If the ratios of all three pairs of corresponding sides are equal, the triangles are similar. For triangles $\triangle ABC$ and $\triangle DEF$: $$ \frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD} $$
• Side-Angle-Side (SAS) Similarity: If the ratio of two pairs of corresponding sides are equal, and the included angles are equal, the triangles are similar. For triangles $\triangle ABC$ and $\triangle DEF$: $$ \frac{AB}{DE} = \frac{AC}{DF} \text{ and } \angle A = \angle D$$
• Angle-Angle (AA) Similarity: If two pairs of corresponding angles are equal, the triangles are similar. (Since the sum of angles in a triangle is 180 degrees, knowing two angles implies knowing the third). For triangles $\triangle ABC$ and $\triangle DEF$: $$ \angle A = \angle D \text{ and } \angle B = \angle E $$

Examples

Here are a couple of examples to illustrate the concept of similarity.

Example-1: Consider two triangles, $\triangle ABC$ and $\triangle DEF$. Suppose $AB = 6$, $BC = 8$, $CA = 10$, $DE = 3$, $EF = 4$, and $FD = 5$.

We can check the ratios of the corresponding sides: $$ \frac{AB}{DE} = \frac{6}{3} = 2, \quad \frac{BC}{EF} = \frac{8}{4} = 2, \quad \frac{CA}{FD} = \frac{10}{5} = 2 $$ Since all ratios are equal, the triangles are similar by SSS similarity.

Example-2: Consider two triangles, $\triangle GHI$ and $\triangle JKL$. Suppose $\angle G = 60^\circ$, $\angle H = 70^\circ$, $\angle J = 60^\circ$, and $\angle K = 70^\circ$.

Since $\angle G = \angle J$ and $\angle H = \angle K$, and using the AA similarity criterion the triangles are similar.

Theorem with Proof

Theorem: If a line is drawn parallel to one side of a triangle to intersect the other two sides at distinct points, then the other two sides are divided in the same ratio.

Proof: Let’s consider a triangle $\triangle ABC$, and let $DE$ be a line segment parallel to $BC$, intersecting $AB$ at $D$ and $AC$ at $E$.

1. Draw perpendiculars from $D$ and $E$ to $BC$, let’s call the points of intersection $F$ and $G$ respectively.
2. The area of $\triangle ADE = \frac{1}{2} \times AD \times h_1$, where $h_1$ is the perpendicular distance from $E$ to $AD$. Similarly, the area of $\triangle BDE = \frac{1}{2} \times BD \times h_1$.
3. Therefore, $\frac{Area(\triangle ADE)}{Area(\triangle BDE)} = \frac{AD}{BD}$.
4. Also, the area of $\triangle ADE = \frac{1}{2} \times AE \times h_2$, where $h_2$ is the perpendicular distance from $D$ to $AE$. And the area of $\triangle CDE = \frac{1}{2} \times CE \times h_2$.
5. Therefore, $\frac{Area(\triangle ADE)}{Area(\triangle CDE)} = \frac{AE}{CE}$.
6. Triangles $\triangle BDE$ and $\triangle CDE$ have the same base $DE$ and are between the same parallel lines $DE$ and $BC$. Therefore, $Area(\triangle BDE) = Area(\triangle CDE)$.
7. Since $Area(\triangle BDE) = Area(\triangle CDE)$, we have $\frac{Area(\triangle ADE)}{Area(\triangle BDE)} = \frac{Area(\triangle ADE)}{Area(\triangle CDE)}$, implying $\frac{AD}{BD} = \frac{AE}{CE}$.
8. Hence, the other two sides are divided in the same ratio.

Common mistakes by students

Common mistakes students make when dealing with similar triangles include:

• Incorrectly identifying corresponding sides and angles: Students often struggle with visually matching up the corresponding parts of similar triangles, especially when the triangles are rotated or flipped.
• Using incorrect similarity criteria: Students might misapply the SSS, SAS, or AA similarity criteria, not correctly verifying all the necessary conditions. For example, mistakenly using SSA.
• Setting up incorrect proportions: Errors can occur when setting up ratios of corresponding sides, leading to incorrect calculations. Students might mix up the order of the sides in the ratios.
• Assuming similarity without proof: Students might assume that two triangles are similar based on appearance or limited information without providing any proof, especially in complex diagrams.

Real Life Application

Similarity of triangles has numerous real-life applications, including:

• Architecture and Engineering: Scale models and blueprints used by architects and engineers rely heavily on the concept of similarity.
• Surveying and Mapping: Surveyors use similar triangles to measure distances that are difficult or impossible to measure directly, such as the height of a building or the width of a river (indirect measurement).
• Photography: The principle of similar triangles is used in photography to determine the size and distance of objects in a photograph.
• Navigation: Similar triangles are used in map reading and in navigation systems.

Fun Fact

The concept of similar triangles was used by the ancient Greek mathematician Thales of Miletus to estimate the height of the pyramids in Egypt. He used the shadows cast by the pyramids and a staff to create similar triangles and calculate the pyramid’s height.

Recommended YouTube Videos for Deeper Understanding

Q.1 Which of the following is NOT a condition for the similarity of two triangles?

Check Solution

Ans: D

The SSA condition (Side-Side-Angle) does not guarantee similarity.

Q.2 In $\triangle ABC$ and $\triangle PQR$, if $\frac{AB}{PQ} = \frac{BC}{QR} = \frac{AC}{PR}$, then the two triangles are:

Check Solution

Ans: B

If the ratios of all three sides are equal, the triangles are similar by SSS. However, we don’t have enough information to say if the sides are equal and the figures are congruent.

Q.3 If $\triangle ABC \sim \triangle DEF$, and the ratio of their corresponding sides is 2:3, what is the ratio of their areas?

Check Solution

Ans: C

The ratio of the areas of similar triangles is the square of the ratio of their corresponding sides.

Q.4 In $\triangle XYZ$, $XY = 6$ cm, $YZ = 8$ cm, and $\angle Y = 60^\circ$. In $\triangle UVW$, $UV = 3$ cm, $VW = 4$ cm, and $\angle V = 60^\circ$. Are the triangles similar? If so, by what condition?

Check Solution

Ans: B

$\frac{XY}{UV} = \frac{6}{3} = 2$ and $\frac{YZ}{VW} = \frac{8}{4} = 2$. Also, the included angles are equal. Therefore, the triangles are similar by SAS.

Q.5 If two triangles have all three angles equal, which of the following is true?

Check Solution

Ans: B

If all angles are equal, the triangles are similar (AAA condition).

Next Topic: Criteria for Similarity of Triangles