A kite is in the shape of a square with a diagonal of

32cm
  and an isosceles triangle of base

8cm
  and sides

6cm
  each is to be made of three different shades as shown in the figure. ____ paper of each shade has been used in it?

Question

A kite is in the shape of a square with a diagonal of

32cm
   and an isosceles triangle of base

8cm
   and sides

6cm
   each is to be made of three different shades as shown in the figure. ____ paper of each shade has been used in it?

Accepted Answer

Let us find the area of part 1 first(i) Area of part 1Let us assume that the triangle

ΔABF
  in different orientation as followsWe know that the all sides of square are equal so, let us assume that

⇒BA=AF=a
  We are given that the length of diagonal as

32cm
   so, we have

⇒BF=32cm
  The Pythagoras theorem is given as

b
    2
   
   =
    a
    2
   
   +
    c
    2

.By using the above theorem to triangle

ΔABF
   we get,

⇒B
    A
    2
   
   +A
    F
    2
   
   =B
    F
    2

By substituting the required values we get

⇒
      a
      2
     
     +
      a
      2
     
     =
      32
      2

⇒2
      a
      2
     
     =1024

⇒a=
      
       512
     
     =16
      2
     
     cm

Let us assume that the area of part 1 as

A
    1

.We know that the formula of area of triangle as

⇒A=
    1
    2
   
   (base)(height)
  By using the above formula to

ΔABF
  we get,

⇒
    A
    1
   
   =
    1
    2
   
   (a)(a)=

a
      2

2

By substituting the required values in above equation we get

⇒
      A
      1
     
     =

(162–
          2
         
         )
        2

2

⇒
      A
      1
     
     =
      
       512
      2
     
     =256c
      m
      2

Therefore the area of part 1 is

256c
    m
    2

Now, let us solve the second part.(i) Area of part 2Let us assume that the area of part 2 as

A
    2

We know that the diagonal of a square divides the square into two triangles of same area that is

⇒
    A
    2
   
   =
    A
    1
   
   =256c
    m
    2

Therefore the area of part 2 is

256c
    m
    2

.(iii) Area of part 3Let us take the triangle

ΔCDE
   and draw the height as shown below.We are given that the dimensions of each triangle are

8cm,6cm,6cm
  .So, from the triangle

ΔCDE
   we have,

⇒CE=CD=6cm

⇒DE=8cm

We know that the altitude of isosceles triangle will be the medianSo, the point ‘H’ is mid – point of ‘DE’So, we can the length of ‘HE’ as

⇒HE=
      
       DE
      2

⇒HE=
      8
      2
     
     cm=4cm

The Pythagoras theorem is given as,

b
    2
   
   =
    a
    2
   
   +
    c
    2

.By using the above theorem to triangle

ΔCHE
   we get,

⇒C
    H
    2
   
   +H
    E
    2
   
   =C
    E
    2

By substituting the required values we get

⇒C
      H
      2
     
     +
      4
      2
     
     =
      8
      2

⇒C
      H
      2
     
     =64−16

⇒CH=
      
       48
     
     =4
      3
     
     cm

We know that the formula of area of triangle as

⇒A=
    1
    2
   
   (base)(height)
  By using the above formula to

ΔCDE
  as

A
    3

,

⇒
    A
    3
   
   =
    1
    2
   
   (DE)(CH)
  By substituting the required values in above equation we get

⇒
      A
      3
     
     =
      1
      2

8

4
        3

⇒
      A
      3
     
     =16
      3
     
     c
      m
      2

Therefore the area of third part is

16
    3
   
   c
    m
    2

.

A kite is in the shape of a square with a diagonal of $32 c m$ and an isosceles triangle of base $8 c m$ and sides $6 c m$ each is to be made of three different shades as shown in the figure. ____ paper of each shade has been used in it?

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