Question

Using a square dot sheet or a graph sheet draw a rectangle and construct a similar figure. Find the perimeter and areas of both and compare their ratios with the ratio of their corresponding sides.

Accepted Answer

Two figures are said to be similar, if they have the same shape, but not necessarily the same size.

We need to draw a rectangle on a graph sheet or a dot sheet and construct the similar figure.

As shown in the graph, two similar rectangles are drawn.

In a rectangle $A B C D$ , $A B = C D = 4 cm$ and $A D = B C = 2 cm$

In a rectangle $P Q R S$ , $P Q = R S = 2 cm$ and $P S = Q R = 1 cm$

We know that, perimeter of a rectangle $= 2 (length + breadth)$

Perimeter of a rectangle $A B C D = 2 (4 + 2) = 2 \times 6 = 12 cm$

Perimeter of a rectangle $P Q R S = 2 (2 + 1) = 2 \times 3 = 6 cm$

Now, area of a rectangle $= length \times breadth$

Area of a rectangle $A B C D = 4 cm \times 2 cm = 8 {cm}^{2}$

Area of a rectangle $P Q R S = 2 cm \times 1 cm = 2 {cm}^{2}$

Comparing the ratio of the perimeters with the ratio of the corresponding sides, we get,

$\frac{Perimeter of rectangle A B C D}{Perimeter of rectangle P Q R S} = \frac{A B}{P Q} = \frac{B C}{Q R} = \frac{C D}{R S} = \frac{A D}{P S}$

$\Rightarrow \frac{12 cm}{6 cm} = \frac{4 cm}{2 cm} = \frac{2 cm}{1 cm} = \frac{4 cm}{2 cm} = \frac{2 cm}{1 cm}$

$\Rightarrow \frac{2}{1} = \frac{2}{1} = \frac{2}{1} = \frac{2}{1}$

Therefore, the ratio of the perimeters of two similar rectangles is equal to the ratio of their corresponding sides.

Comparing the ratio of the areas with the ratio of the corresponding sides, we get,

$\frac{Area of a rectangle A B C D}{Area of a rectangle P Q R S} = \frac{A B}{P Q} = \frac{B C}{Q R} = \frac{C D}{R S} = \frac{A D}{P S}$

$\Rightarrow \frac{8 cm}{2 cm} \neq \frac{4 cm}{2 cm} = \frac{2 cm}{1 cm} = \frac{4 cm}{2 cm} = \frac{2 cm}{1 cm}$

$\Rightarrow \frac{4}{1} = \frac{2}{1} = \frac{2}{1} = \frac{2}{1}$

Therefore, the ratio of the areas of two similar rectangles is not equal to the ratio of their corresponding sides.

Using a square dot sheet or a graph sheet draw a rectangle and construct a similar figure. Find the perimeter and areas of both and compare their ratios with the ratio of their corresponding sides.

Important Questions on Triangles

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