Question

The length of the diagonals of a rhombus shaped field are in 28m and 96 m. Find the length of the fence required for fencing the field?

A full detailed explanation will be appreciated

Answer 1

The Concept

To solve this question, we will need to have good knowledge about the properties of a rhombus and the Pythagoras Theorem. We must remember that:

★ Rhombus is a quadrilateral with four equal sides. This means that if one side of the rhombus is 4 cm, the others will also be 4 cm. Also, the perimeter of a rhombus then would be the same as a square's, which is:

$\boxed{\bf{4\times side}}$

★ Its diagonals are perpendicular bisectors of each other. This means that each diagonal bisects the other at right angles, dividing the rhombus into 4 right-angled triangles.

★ The Pythagoras Theorem states that the square on the hypotenuse of a right angled triangle is equal to the sum of the squares on the sides, like:

$\boxed{\bf{h^{2}=s^{2}+s^{2}}}$

Let's begin!

Answer

We are given:

• the length of one diagonal = 28 m
• the length of the other = 96 m

And we need to find the length of the fence required for fencing the field, basically, the perimeter of the rhombus shaped field. To find the perimeter, we must find the side.

As these two diagonals are perpendicularly bisecting each other, four right-angled triangles are formed, with the side of the rhombus as the hypotenuse. To find the side, we must find the hypotenuse.

Let the rhombus shaped field be rhombus ABCD (as in the attachment), where O is the intersection point of the two diagonals.

Let us assume triangle AOD.

Here, one side = half of one diagonal. Therefore,

• OD = 96/2 = 48 m
• OA = 28/2 = 14 m
• AD = hypotenuse

According to the Pythagoras Theorem,

⇒ OD² + OA² = AD²

⇒ 48² + 14² = AD²

⇒ 2304 + 196 = AD²

⇒ 2500 = AD²

⇒ AD = √2500

⇒ AD = 50

Now, we know that the length of the sides of the rhombus is 50 m.

Perimeter of the rhombus = 4 × side

                                           = 4 × 50

                                           = 200 m

Therefore, the length of the fence required for fencing the field is 200 m.