Question

the length of one of the diagonal of a rhombus is greater than other by 6 cm. If it's area is 360 sq. cm .Find length of diagonals ​

Answer 1

Answer :

To Find :-

The two diagonals of the rhombus.

Given :-

• Area of the Rhombus = 360 cm².

We know :-

$\boxed{\underline{\bf{A = \dfrac{1}{2} \times Product of the diagonals}}}$

Concept :-

Let the first diagonal be x .

So , According to the Question, the other diagonal will be (x + 6) cm.

Thus ,

• First Diagonal = x cm

• Second Diagonal = (x + 6) cm

Now , by using this values in the formula for area of a rhombus , we can find the required value.

Solution :-

Using the formula for area of a rhombus and substituting the values in it , we get :-

$:\implies \bf{A = \dfrac{1}{2} \times Product of the diagonals} \\ \\ \\ :\implies \bf{360 = \dfrac{1}{2} \times x \times (x + 6)} \\ \\ \\ :\implies \bf{360 = \dfrac{1}{2} \times x^{2} + 6x)} \\ \\ \\ :\implies \bf{360 \times 2 = x^{2} + 6x} \\ \\ \\ :\implies \bf{720 = x^{2} + 6x} \\ \\ \\ :\implies \bf{0 = x^{2} + 6x - 720} \\ \\ \\ :\implies \bf{x^{2} + (30 - 24)x - 720} \\ \\ \\ :\implies \bf{x^{2} + 30x - 24x - 720} \\ \\ \\ :\implies \bf{x(x + 30) - 24(x + 30)} \\ \\ \\ :\implies \bf{(x + 30)(x - 24)} \\ \\ \\ \therefore \purple{\bf{(x + 30)(x - 24)}}$

Thus, the value of x is - 30 and 24.

Since, the length can't be in negative, the value of x is 24.

Hence, the Length of first diagonal is 24 cm.and the length of other diagonal is (24 + 6) i.e, 30 cm.

Answer 2

Let first diagonal be x cm

Another Diagonal = (x + 6) cm

$\boxed{\color{#ff99ff}{\rm{A = \dfrac{1}{2} \times Product ~of ~diagonals}}}$

$⇢\sf{360 = \dfrac{1}{2} \times x \times (x + 6)}$

$⇢ \sf{360 = \dfrac{1}{2} \times (x^{2} + 6x)}$

$⇢ \sf{360 \times 2 = x^{2} + 6x}$

$⇢ \sf{720 = x^{2} + 6x}$

$⇢ \sf{0 = x^{2} + 6x - 720}$

$⇢ \sf{x^{2} + (30 - 24)x - 720}$

$⇢ \sf{x^{2} + 30x - 24x - 720}$

$⇢ \sf{x(x + 30) - 24(x + 30)}$

$⇢ \sf{(x + 30)(x - 24)}$

$⇢ {\sf{(x + 30)(x - 24)}}$

$</strong><strong>\</strong><strong>s</strong><strong>f</strong><strong> \therefore~</strong><strong>x = -30 </strong><strong>~</strong><strong>and </strong><strong>~</strong><strong>24</strong><strong>$

The length can't be negative, so value of x = 24.

Length of first diagonal = 24 cm

Length of another diagonal (24 + 6) = 30 cm.