Find the relationship between areas and sides of similar triangles class 10th maths practical?

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If two Triangles are similar then their area is equal to the square of the ratio of their corresponding sides

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If two triangles are similar, then the ratio of the area of both triangles is proportional to  the square of the ratio of their corresponding sides

Answer

Relationship between Areas and Sides of Similar Triangles

Similar triangles are triangles that have the same shape but may differ in size. The corresponding angles of similar triangles are equal, and the corresponding sides are in proportion to each other. In this context, let's discuss the relationship between the areas and sides of similar triangles.

1. Ratio of Areas
The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. Mathematically, if two triangles ABC and PQR are similar, then the ratio of their areas is given as:

Area(ABC) / Area(PQR) = (AB/QR)^2 = (BC/PR)^2 = (AC/PQ)^2

This means that if the ratio of the corresponding sides of two similar triangles is known, we can find the ratio of their areas by squaring this ratio.

2. Relationship between Sides and Areas
The relationship between the sides and areas of similar triangles can be understood using the concept of scale factor. The scale factor between two similar triangles is the ratio of the lengths of their corresponding sides.

If the scale factor between the sides of two similar triangles is 'k', then the ratio of their areas will be 'k^2'. This implies that if the corresponding sides of two similar triangles are in the ratio of 'k', then the ratio of their areas will be 'k^2'.

3. Example
Let's consider an example to understand this relationship better. Suppose we have two similar triangles ABC and DEF, where the ratio of the corresponding sides is 3:5.

Given:
AB/DE = BC/EF = AC/DF = 3/5

We can find the ratio of their areas as:

Area(ABC) / Area(DEF) = (3/5)^2 = 9/25

This implies that the area of triangle ABC is 9/25 times the area of triangle DEF.

Conclusion
In conclusion, the relationship between the areas and sides of similar triangles can be summarized as follows:

- The ratio of the areas of similar triangles is equal to the square of the ratio of their corresponding sides.
- The ratio of the areas is equal to the square of the scale factor between the sides of similar triangles.
- If the corresponding sides of two similar triangles are in the ratio of 'k', then the ratio of their areas will be 'k^2'.

Understanding this relationship helps in solving problems related to similar triangles, such as finding missing side lengths or areas.

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