the areas of two similar polygons are 64 sq units and 100 sq units respectively. if a side of the larger polygon is 36 units, find the corresponding side of the smaller polygon.

To solve this problem, we use the fact that for two similar polygons, the ratio of their areas is equal to the square of the ratio of their corresponding sides.

Given:
- Area of the smaller polygon = 64 square units
- Area of the larger polygon = 100 square units
- A side of the larger polygon = 36 units

Let \( x \) represent the corresponding side of the smaller polygon.

The relationship between the areas and sides is given by:

\[
\frac{\text{Area of smaller polygon}}{\text{Area of larger polygon}} = \left(\frac{\text{Side of smaller polygon}}{\text{Side of larger polygon}}\right)^2
\]

Substituting the given values:

\[
\frac{64}{100} = \left(\frac{x}{36}\right)^2
\]

Simplify:

\[
\frac{8}{10} = \left(\frac{x}{36}\right)^2
\]

\[
\frac{4}{5} = \left(\frac{x}{36}\right)^2
\]

Now, take the square root of both sides:

\[
\sqrt{\frac{4}{5}} = \frac{x}{36}
\]

\[
\frac{2}{\sqrt{5}} = \frac{x}{36}
\]

Now, solve for \( x \):

\[
x = 36 \times \frac{2}{\sqrt{5}} = \frac{72}{\sqrt{5}}
\]

To rationalize the denominator:

\[
x = \frac{72}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{72\sqrt{5}}{5}
\]

Thus, the corresponding side of the smaller polygon is:

\[
x \approx 32.2 \, \text{units}
\]

Would you like more details or clarifications?

Here are 5 questions to further explore this concept:
1. How does the ratio of perimeters relate to the ratio of areas in similar polygons?
2. Can this concept of area and side ratios be applied to 3D shapes like similar solids?
3. What is the effect on side lengths if the ratio of the areas is reversed?
4. How do you find the scale factor between two similar polygons based on their areas?
5. What happens to the ratio of areas if both side lengths double?

**Tip:** In similar polygons, the ratio of areas is always the square of the ratio of corresponding sides.

The areas of two similar polygons are 64 sq units and 100 sq units respectively. If a side of the larger polygon is 36 units, find the corresponding side of the smaller polygon.

Learn how to find the corresponding side of a smaller polygon when the areas of two similar polygons are 64 sq units and 100 sq units, and the side of the larger polygon is 36 units. This problem covers geometry concepts, including the ratio of areas and square root relationships. Suitable for students in Grades 8-10.

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