Question

The area of a rhombus is 600 sq.units and the diagonals are in ratio 3:4. Find the diagonals and one side.

Answer 1

Answer:

let the diagonals be 3x and 4x

area of rhombus =1/2 ×diagnal 1 ×diagnal 2

600=1/2×3x×4x

600=6x²

x²=600/6=100

x=50

diagnal1 =3x=3×50=150

diagnal 2=4x=4×50=200

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Answer 2

Answer:

• Diagonal 1 of rhombus = 30 units.
• Diagonal 2 of rhombus = 40 units.
• Length of side of rhombus = 25 units.

Step-by-step explanation:

Given :-

• The area of the rhombus is 600 sq.units.
• The ratio of the diagonals of the rhombus is 3 : 4.

To find :-

• Diagonals of the rectangle.
• Length of side of the rhombus.

Solution :-

To find the length of the diagonal of the rhombus, let us assume the length of the first diagonal be 3x units and the length of second diagonal be 4x units, since the ratio of diagonals of rhombus is 3 : 4. So, by using the formula of area of rhombus we will find out diagonals.

$\sf \longrightarrow {Area\ of\ rhombus\ =\ \dfrac{1}{2} \times d_1 \times d_2}$

• On substituting the values,

$\sf \longrightarrow {600\ =\ \dfrac{1}{2} \times 3x \times 4x}$

$\sf \longrightarrow {600\ =\ \dfrac{1}{\cancel 2} \times \cancel{12x^2}}$

$\sf \longrightarrow {600\ =\ 6x^2}$

$\sf \longrightarrow {\dfrac{\cancel{600}}{\cancel 6}\ =\ x^2}$

$\sf \longrightarrow {100\ =\ x^2}$

$\sf \longrightarrow {\sqrt{100}\ =\ x}$

$\sf \longrightarrow {10\ =\ x}$

$\sf \longrightarrow {\underline{x\ =\ 10}\ \bigstar}$

We get the value of x as 10. So now we will find out the diagonals of rhombus.

• Diagonal 1 : 3x = 3(10) = 3 × 10 = 30 units.
• Diagonal 2 : 4x = 4(10) = 4 × 10 = 40 units.

When the diagonals intersects each other in the rhombus, they form four right angled triangles. So, according to this our required parameters will be :

• Hypotenuse : Side.
• Perpendicular : 1/2 × Diagonal 2 = 1/2 × 40 = 20 units.
• Base : 1/2 × Diagonal 1 = 1/2 × 30 = 15 units.

So now by using Pythagoras Theorem we will find out length of the side of the rhombus.

$\sf \longrightarrow {(Hypotenuse)^2\ =\ (Perpendicular)^2\ +\ (Base)^2}$

• On substituting the values,

$\sf \longrightarrow {(Hypotenuse)^2\ =\ (20)^2\ +\ (15)^2}$

$\sf \longrightarrow {(Hypotenuse)^2\ =\ 400\ +\ 225}$

$\sf \longrightarrow {(Hypotenuse)^2\ =\ 625}$

$\sf \longrightarrow {Hypotenuse\ =\ \sqrt{625}}$

$\sf \longrightarrow {\underline{Hypotenuse\ =\ 25}\ \bigstar}$

∴ Hence, length of the side of the rhombus is 25 units.