Question

Find the area of the rhombus whose one of the equal sides is 100 m and one of the diagonals is 160 m​

Answer 1

The area of the rhombus is 9600 m².

Step-by-step explanation:

In the the question, it's been stated that the side of a rhombus is 100 m and one of its diagonal equals to 160 m. Here, we've been asked to determine the area of the rhombus.

$\qquad \underline{ \boxed{ \sf \pmb{Area = \dfrac{1}{2} \times d_{1} \times d_{2} \: (sq.units)}}}$

In order to find the value of second diagonal, we will need to use the Pythagorean Theorem. You may think why. Yeah, here is the solution!

In a right-angled triangle, we have: a perpendicular (a) , a base (b) and a hypotenuse (c). And in a rhombus, the diagonals divide it into four right angles, where:

• The perpendicular equals to the half of the first diagonal.
• The base equals to the half of the second diagonal.
• The hypotenuse equals to the length of side of the rhombus.

So, as to determine the measure of 2nd diagonal (2 × base), we must use Pythagorean Theorem.

Applying Pythagorean Theorem:

$\twoheadrightarrow \quad{ \sf{ {(a)}^{2} + {(b)}^{2} = {(c)}^{2} }}$

$\twoheadrightarrow \quad{ \sf{ { \bigg(\dfrac{160}{2} } \bigg)^{2} + {(b)}^{2} = {(100)}^{2} }}$

$\twoheadrightarrow \quad{ \sf{ {(80)}^{2} + {(b)}^{2} = {(100)}^{2} }}$

$\twoheadrightarrow \quad{ \sf{{(b)}^{2} = {(100)}^{2} - {(80)}^{2} }}$

$\twoheadrightarrow \quad{ \sf{{(b)}^{2} = 10000 - 6400 }}$

$\twoheadrightarrow \quad{ \sf{{(b)}^{2} =3600 }}$

$\twoheadrightarrow \quad{ \sf{b = \sqrt{3600 }}}$

$\twoheadrightarrow \quad{ \sf{Base = 60 \: m}}$

As we know,

$\twoheadrightarrow \quad{ \sf{2nd \: Diagonal = 2(Base)}}$

The measure of the second diagonal equals to -

$\twoheadrightarrow \quad{ \sf{2(60)}}$

$\twoheadrightarrow \quad{ \sf \pmb{120 \: m}}$

Since the length of the second diagonal is 120 m, the area of the rhombus equals to -

Using the formula:

$\twoheadrightarrow \quad{ \sf{ \dfrac{1}{2} \times d_{1} \times d_{2}}}$

$\twoheadrightarrow \quad{ \sf{ \dfrac{1}{2} \times 160 \times 120}}$

$\twoheadrightarrow \quad{ \sf{ \dfrac{1}{2} \times 19200}}$

$\twoheadrightarrow \quad{ \sf{ \dfrac{1 \times 19200}{2}}}$

$\twoheadrightarrow \quad{ \sf{ \dfrac{19200}{2}}}$

$\twoheadrightarrow \quad{ \sf{ \dfrac{9600}{1}}}$

$\twoheadrightarrow \quad \underline{ \boxed{ \frak{ \pmb {\red{9600 \: {m}^{2} }}}}}$

∴ The area of the rhombus when one of its diagonal and its side are 160 m and 100 m, respectively equals to 9600 m².