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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

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?

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 cm²

Since the diagonal divides the square into two equal triangles. Therefore, Area of ∆ABD = Area of ∆CBD = 256 cm²

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 cm²

Thus, the area of the paper used to make region I = 256 cm², region II = 256 cm², and region III = 17.92 cm².

☛ 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 cm², shade II = 256 cm², and shade III = 17.92 cm² has been used in it.

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