What is SSS Similarity Criterion?

S3-SA2-0056

Grade Level:

Class 6

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

The SSS (Side-Side-Side) Similarity Criterion is a rule that helps us decide if two triangles are similar just by looking at the lengths of their sides. If the ratios of the lengths of all three corresponding sides of two triangles are equal, then the triangles are similar.

Simple Example

Quick Example

Imagine you have two photographs of the same monument, like the Taj Mahal. One photo is a small passport size, and the other is a larger poster size. Even though they are different sizes, they look exactly alike in shape. If you measure the sides of the Taj Mahal in both photos, the ratio of corresponding sides (like the ratio of the width of the main dome in the small photo to the width of the main dome in the large photo) will be the same for all parts. This means the two photos are similar, based on SSS.

Worked Example

Step-by-Step

Let's check if Triangle ABC and Triangle PQR are similar using SSS criterion.

Triangle ABC has sides: AB = 6 cm, BC = 9 cm, AC = 12 cm.
Triangle PQR has sides: PQ = 2 cm, QR = 3 cm, PR = 4 cm.

Step 1: Identify corresponding sides. The smallest side of ABC is AB (6 cm) and the smallest side of PQR is PQ (2 cm). So, AB corresponds to PQ.
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Step 2: Find the ratio of the first pair of corresponding sides. Ratio 1 = AB / PQ = 6 / 2 = 3.
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Step 3: Identify the next pair of corresponding sides. The medium side of ABC is BC (9 cm) and the medium side of PQR is QR (3 cm). So, BC corresponds to QR.
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Step 4: Find the ratio of the second pair of corresponding sides. Ratio 2 = BC / QR = 9 / 3 = 3.
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Step 5: Identify the largest pair of corresponding sides. The largest side of ABC is AC (12 cm) and the largest side of PQR is PR (4 cm). So, AC corresponds to PR.
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Step 6: Find the ratio of the third pair of corresponding sides. Ratio 3 = AC / PR = 12 / 4 = 3.
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Step 7: Compare the ratios. All three ratios are equal (Ratio 1 = 3, Ratio 2 = 3, Ratio 3 = 3).
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Answer: Since the ratios of all three corresponding sides are equal, Triangle ABC is similar to Triangle PQR by the SSS Similarity Criterion.

Why It Matters

Understanding similarity is crucial in fields like engineering and architecture, where models are scaled versions of real buildings. In computer graphics and animation, artists use similarity to resize images and objects without distorting their shape. Even in data science, recognizing similar patterns helps in analyzing complex information.

Common Mistakes

MISTAKE: Not matching corresponding sides correctly (e.g., comparing the smallest side of one triangle with the largest side of another). | CORRECTION: Always match the smallest side with the smallest, the medium with the medium, and the largest with the largest side of the other triangle.

MISTAKE: Calculating the ratio incorrectly (e.g., dividing side 1 by side 2 for one pair, and side 2 by side 1 for another pair). | CORRECTION: Be consistent. If you divide the side of the first triangle by the side of the second triangle for one pair, do the same for all other pairs.

MISTAKE: Concluding similarity even if only one or two ratios are equal. | CORRECTION: For SSS similarity, ALL THREE ratios of corresponding sides MUST be equal.

Practice Questions

Try It Yourself

QUESTION: Triangle XYZ has sides XY=5, YZ=10, XZ=15. Triangle LMN has sides LM=1, MN=2, LN=3. Are they similar by SSS? | ANSWER: Yes, they are similar. Ratios are 5/1 = 5, 10/2 = 5, 15/3 = 5.

QUESTION: Triangle PQR has sides PQ=4, QR=6, RP=8. Triangle DEF has sides DE=6, EF=9, FD=12. Are they similar by SSS? | ANSWER: Yes, they are similar. Ratios are 4/6 = 2/3, 6/9 = 2/3, 8/12 = 2/3.

QUESTION: Two triangles have sides (10, 12, 14) and (15, 18, 21). Are they similar? If yes, what is the ratio of their corresponding sides? | ANSWER: Yes, they are similar. The ratio of corresponding sides is 10/15 = 2/3, 12/18 = 2/3, 14/21 = 2/3. So, the ratio is 2/3.

MCQ

Quick Quiz

Which of the following conditions must be true for two triangles to be similar by SSS criterion?

All three angles of one triangle are equal to the corresponding angles of the other triangle.

The ratio of at least two pairs of corresponding sides is equal.

The ratio of all three pairs of corresponding sides is equal.

One angle and two sides are equal in both triangles.

The Correct Answer Is:

The SSS Similarity Criterion specifically states that for two triangles to be similar, the ratios of ALL THREE pairs of their corresponding sides must be equal. Option A describes AAA similarity, and B and D are incorrect conditions for SSS.

Real World Connection

In the Real World

When architects design buildings in India, they often create small-scale models. These models are similar to the actual building. For example, if a builder wants to construct a new apartment complex in Mumbai, they might first make a miniature model. They use SSS similarity to ensure that every part of the model is proportionally correct to the actual building, allowing them to visualize the final structure before construction begins.

Key Vocabulary

Key Terms

SIMILAR TRIANGLES: Triangles that have the same shape but can be different sizes. | CORRESPONDING SIDES: Sides that are in the same relative position in two different triangles. | RATIO: A comparison of two numbers by division. | CRITERION: A rule or standard by which something can be judged or decided.

What's Next

What to Learn Next

Great job learning about SSS Similarity! Next, you should explore the SAS (Side-Angle-Side) Similarity Criterion and AAA (Angle-Angle-Angle) Similarity Criterion. These are other important rules that will help you identify similar triangles using different combinations of sides and angles.