**Area and perimeter of similar figures :**

Two figures that have the same shape are said to be similar. When two figures are similar, the ratios of the lengths of their corresponding sides are equal. To determine if the triangles below are similar, compare their corresponding sides.

length of side AB/length of side AD = Length of side EF/Length of side EH

(4/2) = (8/4)

2 : 1 = 2 : 1

**Perimeter :**

If two similar triangles have a scale factor of a : b, then the ratio of their perimeters is a : b.

**Area :**

If two similar triangles have a scale factor of a : b, then the ratio of their areas is a^{2} : b^{2}.

Let us see some examples based on the above concept.

**Example 1 :**

Find the missing length of following similar shapes

**Solution :**

Since the above rectangles ABCD and EFGH are similar, the ratio of the corresponding sides will be in the same ratio.

Length of side AB/Length of side AD = Length of EF/Length of EH

4/6 = a/12

4 (12) = 6a

a = 48/6

a = 8

Hence the missing length of the rectangle EFGH is 8 cm.

**Example 2 :**

Find the missing length of following similar shapes

**Solution :**

Since the above rectangles ABCD and EFGH are similar, the ratio of the corresponding sides will be in the same ratio.

Length of side AB/Length of side AD = Length of EF/Length of EH

b/1 = 7.5/3

b (3) = 7.5 (1)

b = 7.5/3

b = 2.5

Hence the missing length of the rectangle EFGH is 2.5 cm.

**Example 3 :**

The perimeters of two similar triangles is in the ratio 3 : 4. The sum of their areas is 75 cm^{2}. Find the area of each triangle.

**Solution :**

Since the perimeters of two similar triangles is in the ratio 3 : 4,

Let "3x" be the side length of first triangle

Let "4x" be the side length of second triangle

Area of 1st triangle/Area of 2nd triangle = (3x/4x)²

= 9x²/16x²

The sum of their areas = 75 cm^{2}

9x²+ 16x² = 75

25x² = 75

x² = 75/25 = 3

Area of first triangle = 9x² = 3(3) = 9 cm²

Area of second triangle = 16x² = 16(3) = 48 cm²

**Example 4 :**

The areas of two similar triangles are 45 cm^{2} and 80 cm^{2}. The sum of their perimeters is 35 cm. Find the perimeter of each triangle.

let the scale factor of the two similar triangles be a : b.

Area of 1st triangle/Area of 2nd triangle = 45/80

*a* : *b* is the reduced form of the scale factor. 3 : 4 is then the reduced form of the comparison of the perimeters.

(9/16) = (a/b)^{2}

(a/b) = 3/4

Perimeter of 1st triangle = 3x

Perimeter of 2nd triangle = 4x

Sum of the perimeters of the 1st and 2nd triangle = 35

3x + 4x = 35

7x = 35 ==> x = 5

Perimeter of 1st triangle = 3x = 3(5) = 15 cm

Perimeter of 2nd triangle = 4x = 4(5) = 20 cm

After having gone through the stuff given above, we hope that the students would have understood "Area and perimeter of similar figures".

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