Question

The diagonals of a rhombus are in the ratio 3:4. If its perimeter is 40 cm, find the lengths of the sides and diagonals of the rhombus

Answer 1

Let the length of the side of the rhombus be x
Since the sides of a rhombus are equal,
    4x=40
      x=40/4
        =10 cm

let the diagonals d1 and d2 of a rhombus be 3y and 4y respectively.( since the ratio of the diagonals are given as 3:4)

Diagonals of a rhombus are perpendicular bisectors.
Therefore a rhombus can be divided into four right triangles.

Considering a triangle from the rhombus,

By Pythagoras theorem,
  x^2     =     (d1/2)^2     +  (d2/2)^2
 10^2    =     (3y/2)^2     +  (4y/2)^2
 100     =      (9(y^2))/4  + (16(y^2))/4
 400     =      (9+16)  (y^2)
400      =      25 (y^2)
y^2      =      400/25
y^2      =      16
y          =      4

therefore the diagonals of the rhombus are:
d1 = 3y = 3 *4 = 12 cm
d2 = 4y = 4 *4 = 16 cm

the length of the side of the rhombus is 10 cm and the diagonals d1 and d2 are 12 cm and 16 cm respectively.

Answer 2

Let the length of the side of the rhombus be x

Since the sides of a rhombus are equal,

    4x=40

      x=40/4

        =10 cm

let the diagonals d1 and d2 of a rhombus be 3y and 4y respectively.( since the ratio of the diagonals are given as 3:4)

Diagonals of a rhombus are perpendicular bisectors.

Therefore a rhombus can be divided into four right triangles.

Considering a triangle from the rhombus,

By Pythagoras theorem,

  x^2     =     (d1/2)^2     +  (d2/2)^2

 10^2    =     (3y/2)^2     +  (4y/2)^2

 100     =      (9(y^2))/4  + (16(y^2))/4

 400     =      (9+16)  (y^2)

400      =      25 (y^2)

y^2      =      400/25

y^2      =      16

y          =      4

therefore the diagonals of the rhombus are:

d1 = 3y = 3 *4 = 12 cm

d2 = 4y = 4 *4 = 16 cm

the length of the side of the rhombus is 10 cm and the diagonals d1 and d2 are 12 cm and 16 cm

Step-by-step explanation: