Question

The length of the diagonals of a rhombus are in the ratio 3:4. If its perimeter is 80 cm .

Find the length of the diagonals.
​

Answer 1

Given:

• The ratio of the length of diagonals of a rhombus are 3:4
• The perimeter is 80 cm

To find:

• The length of the diagonals of the rhombus

The rhombus in the attached image is drawn according to the question,

Now,

ABCD is a rhombus, O is the intersecting point of diagonals, AC and BD are diagonals.

Let AC = 3x cm

And BD = 4x cm

We know that,

• The diagonals of a rhombus bisect each other at 90°.
• The perimeter of the rhombus is 4*side.

So,

• AO = OC = 1/2AC = 3x/2
• OD = OB = 1/2BD = 4x/2 = 2x

The perimeter of the rhombus = 4 * side

⇒ 80 = 4 * side

⇒ 80 ÷ 4 = side

⇒ 20 cm = side

• Now, there are four right-angled triangles in the rhombus.

ΔAOB, ΔBOC, ΔCOD, ΔAOD

In ΔAOB,

[Using Pythagoras Theorem]

→ (AO)² + (OB)² = (20)²

→ (3x/2)² + (2x)² = 400

→ (9x²/4) + 4x² = 400

→ (9x²/4) + 4x² = 400

[Taking LCM]

→ (9x²+16x²)/4 = 400

→ (25x²)/4 = 400

[Taking square root in both RHS and LHS]

→ 5x/2 = 20

→ 5x = 20*2

→ 5x = 40

→ x = 40/5

∴ x = 8

Now,

One diagonal is 3x = 3*8 = 24 cm

Another diagonal is 4x = 4*8 = 32 cm

Answer 2

$\Large\bf{\color{indigo}GiVeN,}$

• The length of diagonals of a rhombus are in ratio 3 : 4.

• It's perimeter is 80 cm.

$\bf\pink{Let,}$

• The length of one diagonal is 3X.

=》 d₁ = 3X

• And length of other diagonal is 4X.

=》 d₂ = 4X

$\bf\blue{We\:have,}$

$\red\bigstar\:\:\bf{\color{peru}Perimeter\:=\:4\times{Side}\:}$

$:\implies\:\:\bf{80\:=\:4\times{Side}\:}$

$:\implies\:\:\bf{Side\:=\:\dfrac{80}{4}\:}$

$:\implies\:\:\bf\green{Side\:=\:20\:cm\:}$

$\bf\purple{We\:know\:that,}$

❶ In rhombus all sides are equal and two diagonals are intersect at the mid point of each other.

❷ At the intersection point two diagonals makes 90° angle between them.

$\bf\red{So,}$

$\pink\bigstar\:\:\bf{\color{coral}(Side)^2\:=\:\Big(\dfrac{d_1}{2}\Big)^2\:+\:\Big(\dfrac{d_2}{2}\Big)^2\:}$

$:\implies\:\:\bf{(20)^2\:=\:\Big(\dfrac{3X}{2}\Big)^2\:+\:\Big(\dfrac{4X}{2}\Big)^2\:}$

$:\implies\:\:\bf{400\:=\:\dfrac{9X^2}{4}\:+\:\dfrac{16X^2}{4}\:}$

$:\implies\:\:\bf{400\:=\:\dfrac{9X^2\:+\:16X^2}{4}\:}$

$:\implies\:\:\bf{400\times{4}\:=\:25X^2\:}$

$:\implies\:\:\bf{1600\:=\:25X^2\:}$

$:\implies\:\:\bf{X^2\:=\:\dfrac{1600}{25}}$

$:\implies\:\:\bf{X\:=\:\sqrt{\dfrac{1600}{25}}\:}$

$:\implies\:\:\bf{X\:=\:\dfrac{40}{5}\:}$

$:\implies\:\:\bf{\color{peru}X\:=\:8\:cm\:}$

________________

$\bf\orange{Hence,}$

=》 d₁ = 3 × 8 = 24 cm

=》 d₂ = 4 × 8 = 32 cm

$\Large\bold\therefore$ The length of the diagonals are 24 cm & 32 cm.