# A kite in the shape of a square with a diagonal 32 cm and an isosceles triangle of base 8 cm and sides 6 cm each is to be made of three different shades as shown in Fig. 12.17. How much paper of each shade has been used in it?

**Solution:**

We divide the kite into three different triangles, and by using Heron’s formula, we can calculate the area of triangle.

Heron's formula for the area of a triangle = √s(s - a)(s - b)(s - c)

Where a, b, and c are the sides of the triangle, and

s = Semi-perimeter = Half the Perimeter of the triangle = (a + b + c)/2

We know that the diagonals of a square are perpendicular bisectors of each other.

Given diagonal BD = AC = 32 cm, then OA = 1/2 AC = 16 cm.

So square ABCD is divided into two isosceles triangles ABD and CBD of base 32 cm and height 16 cm.

Area of ∆ABD = 1/2 × base × height

= (32 × 16)/2

= 256 cm^{2}

Since the diagonal divides the square into two equal triangles. Therefore, Area of ∆ABD = Area of ∆CBD = 256 cm^{2}

Now, for ∆CEF

Semi Perimeter(s) = (a + b + c)/2

s = (6 + 6 + 8)/2

s = 20/2

s = 10 cm

By using Heron’s formula,

Area of ∆CEF = √s(s - a)(s - b)(s - c)

= √10(10 - 6)(10 - 6)(10 - 8)

= √10 × 4 × 4 × 2

= 8√5

= 8 × 2.24

= 17.92 cm^{2}

Thus, the area of the paper used to make region I = 256 cm^{2}, region II = 256 cm^{2}, and region III = 17.92 cm^{2}.

**☛ Check: **NCERT Solutions Class 9 Maths Chapter 12

**Video Solution:**

## A kite in the shape of a square with a diagonal 32 cm and an isosceles triangle of base 8 cm and sides 6 cm each is to be made of three different shades as shown in Fig. 12.17. How much paper of each shade has been used in it?

Class 9 Maths NCERT Solutions Chapter 12 Exercise 12.2 Question 7

**Summary:**

It is given that there is a kite in the shape of a square with a diagonal 32 cm and an isosceles triangle of base 8 cm and sides 6 cm each is to be made of three different shades. We have found that the area of the paper of shade I = 256 cm^{2}, shade II = 256 cm^{2}, and shade III = 17.92 cm^{2} has been used in it.

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