What is the Similarity of Polygons?

S3-SA2-0324

Grade Level:

Class 6

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

Definition

What is it?

The similarity of polygons means that two polygons have the same shape, but not necessarily the same size. Imagine taking a photo and then zooming in or out – the zoomed picture is similar to the original because its shape hasn't changed, only its size.

Simple Example

Quick Example

Think about two different-sized photographs of the Gateway of India. Both photos show the same monument and have the same rectangular shape, but one might be a small wallet-sized photo and the other a large poster. They are similar because their angles are the same, and their sides are in proportion.

Worked Example

Step-by-Step

Let's check if two rectangles, Rectangle A (sides 4 cm and 6 cm) and Rectangle B (sides 8 cm and 12 cm), are similar.

Step 1: Identify corresponding angles. All angles in a rectangle are 90 degrees. So, the corresponding angles of Rectangle A and Rectangle B are equal.
---Step 2: Identify corresponding sides. The shorter side of Rectangle A is 4 cm, and the shorter side of Rectangle B is 8 cm. The longer side of Rectangle A is 6 cm, and the longer side of Rectangle B is 12 cm.
---Step 3: Calculate the ratio of corresponding sides. Ratio of shorter sides = 8 cm / 4 cm = 2. Ratio of longer sides = 12 cm / 6 cm = 2.
---Step 4: Compare the ratios. Since both ratios are equal (2), the corresponding sides are in proportion.
---Answer: Yes, Rectangle A and Rectangle B are similar polygons because their corresponding angles are equal and their corresponding sides are in proportion.

Why It Matters

Understanding similarity is crucial in fields like architecture and engineering, where designers create scaled models before building large structures. In computer graphics, it helps in resizing images or objects without distorting them, used by game developers and animators. It's also fundamental in AI for pattern recognition.

Common Mistakes

MISTAKE: Thinking that if two polygons have the same number of sides, they are always similar. | CORRECTION: Polygons must have both equal corresponding angles AND proportional corresponding sides to be similar.

MISTAKE: Only checking if the angles are equal, ignoring the side ratios. | CORRECTION: Both conditions (equal angles AND proportional sides) must be met for polygons to be similar.

MISTAKE: Comparing non-corresponding sides when checking ratios. | CORRECTION: Always match the smallest side of one polygon with the smallest side of the other, and so on, to find the correct corresponding sides.

Practice Questions

Try It Yourself

QUESTION: Are two squares always similar to each other? | ANSWER: Yes, because all angles in a square are 90 degrees (equal), and all sides in a square are equal, so their ratios will always be the same.

QUESTION: A triangle has sides 3 cm, 4 cm, 5 cm. Another triangle has sides 6 cm, 8 cm, 10 cm. If their angles are also equal, are they similar? | ANSWER: Yes. Ratios of corresponding sides are 6/3 = 2, 8/4 = 2, 10/5 = 2. Since angles are equal and sides are proportional, they are similar.

QUESTION: Rectangle P has sides 5m and 10m. Rectangle Q has sides 6m and 15m. Are they similar? Explain why. | ANSWER: No. The ratios of corresponding sides are 6/5 = 1.2 and 15/10 = 1.5. Since the ratios are not equal (1.2 is not equal to 1.5), the rectangles are not similar, even though their angles are all 90 degrees.

MCQ

Quick Quiz

Which of these is a condition for two polygons to be similar?

They must have the same area.

Their corresponding angles must be equal.

Their corresponding sides must be equal.

They must have the same perimeter.

The Correct Answer Is:

For polygons to be similar, their corresponding angles must be equal, and their corresponding sides must be in proportion. Options A, C, and D are not required conditions for similarity.

Real World Connection

In the Real World

When you use Google Maps or any navigation app on your phone, the map displayed is a smaller, similar version of the actual geographical area. The shapes of roads and buildings are maintained, but everything is scaled down. This application of similarity helps you navigate effectively.

Key Vocabulary

Key Terms

POLYGON: A closed 2D shape with straight sides. | CORRESPONDING ANGLES: Angles that are in the same relative position in two different polygons. | CORRESPONDING SIDES: Sides that are in the same relative position in two different polygons. | PROPORTIONAL: When two ratios are equal.

What's Next

What to Learn Next

Now that you understand similarity, you can explore 'Congruence of Polygons.' Congruence is a special type of similarity where not only are the shapes the same, but the sizes are identical too! It's an exciting next step in understanding geometric relationships.