Question

area of a rhombus is 75 sqcm If one of the diagnol is 5cm longer then the other find the length of both the diagnols (answer is 10 and 15 cm)​

Answer 1

Answer :-

Lengths of both diagonals are 10 cm and 15 cm.

Solution :-

Let the length of the other diagonal be 'x' cm

Length of the one of the diagonal = 5 cm longer than the other = (x + 5) cm

Given

Area of the Rhombus = 75 cm²

⇒ d1 * d2/2 = 75

⇒ x * (x + 5)/2 = 75

⇒ x * (x + 5) = 75 * 2

⇒ x² + 5x = 150

⇒ x² + 5x = 150

⇒ x² + 5x - 150 = 0

⇒ x² + 15x - 10x - 150 = 0

⇒ x(x + 15) - 10(x + 15) = 0

⇒ (x - 10)(x + 15) = 0

⇒ x - 10 = 0 or x + 15 = 0

⇒ x = 10 or x = - 15

Length of the diagonal cannot be negative.

So x = 10 cm

Length of other diagonal = x = 10 cm

Length of one of the diagonal = (x + 5) = (10 + 5) = 15 cm

Therefore the lengths of both diagonals are 10 cm and 15 cm.

Answer 2

$\large{\mathfrak{\underline{\underline{Answer:-}}}}$

${\mathfrak{\underline{\underline{step-by-step-Explanation:-}}}}$

Area = 75 cm²

let the one diagnonal be a .

So,

second diagonal will be (a + 5)

__________________________

$\huge{\bf{\boxed{\boxed{{\frac{D_{1} \: {\times} D_{2}}{2}} \: = \: Area}}}}$

________________[Put values]

⟹ a * (a + 5)/2 = 75

⟹ a * (a + 5) = 75*2

⟹ a (a + 5) = 150

⇒ a² + 5a = 150 = 0

⟹ a² + 5a -150 = 0

We get a quadratic equation, Solve it

by splitting the middle term method.

⟹ a² + 15a - 10a -150 = 0

⟹ a(a + 15) - 10 (a + 15) = 0

⟹ (a - 10) (a + 15) = 0

⟹ a = 10

⟹ a = -15

Diagonal can't be negative so, we will use positive one.

a = 10 cm

$\large{\sf{\boxed{\boxed{Diagonal_{1} \: = \: 10cm}}}}$

___________________________

Second diagonal is

a + 10

⟹ 10 + 5

⟹ 15 cm

$\large{\sf{\boxed{\boxed{Diagonal_{2} \: = \: 15cm}}}}$