Question

find the area of rhombus whose one side measures 5cm and one diagonal as 8cm​

Answer 1

$\blue{\bold{\underline{\underline{Answer:}}}}$

$\green{\tt{\therefore{Area\:of\:rhombus=24\:cm^{2}}}}$

$\orange{\bold{\underline{\underline{Step-by-step\:explanation:}}}}$

$\green{\underline \bold{Given: }} \\ \tt: \implies Side \: of \: rhombus \: ABCD= 5 \: cm \\ \\ \tt: \implies Diagonal \: of \: rhombus \: ABCD= 8 \: cm \\ \\ \red{\underline \bold{To \: Find: }} \\ \tt: \implies Area \: of \: rhombus = ?$

• According to given question :

$\tt \circ \:AB = 5 \: cm \\ \\ \tt \circ \: AC = 8 \: cm \\ \\ \tt \circ \: AO = \frac{1}{2} AC = \frac{1}{2} \times 8 = 4 \: cm \\ \\ \bold{As \: we \: know \: that} \\ \tt: \implies {h}^{2} = {p}^{2} + {b}^{2} \: \: \: \: \: \: (By \: phythagoras \: theorem) \\ \\ \tt: \implies {AB}^{2} = {AO}^{2} + {BO}^{2} \\ \\ \tt: \implies {5}^{2} = {4}^{2} + {BO}^{2} \\ \\ \tt: \implies 25 = 16 + {BO}^{2} \\ \\ \tt: \implies BO = \sqrt{9} \\ \\ \green{\tt: \implies BO = 3 \: cm} \\ \\ \tt: \implies BD = 2BO \\ \\ \green{\tt: \implies BD = 2 \times 3 = 6 \: cm} \\ \\ \bold{As \: we \: know \: that} \\ \tt: \implies Area \: of \: rhombus = \frac{1}{2} \times D_{1} \times D_{2} \\ \\ \tt: \implies Area \: of \: rhombus = \frac{1}{2} \times 8 \times 6 \\ \\ \green{\tt: \implies Area \: of \: rhombus =24 \: {cm}^{2}}$

Answer 2

Answer:-

The area of rhombus is 24 cm².

Given :

(i) Side of Rhombus = 5cm

(ii) Length of diagonal =8cm

To Find:-

Area of Rhombus = ?

Solution :-

Refer to the figure on the attachment.

As we know that the Rhombus has all sides equal So

AB = BC = CD = AD = 5cm.

And AC = 8cm.

AC = OA + OC = 8cm

So OA = OC = 4cm.

Now, consider ∆AOB,

AB = 5 cm

OA = 4 cm

And as diagonals bisects each other at right angles in a rhombus. So ∆AOB is right angled triangle.

Now by using Pythagoras theorem,

=> OB² + OA² = AB²

=> OB² + 4² = 5²

=> OB² = 5² - 4²

=> OB² = 25 -16

=> OB = 9

=> OB= √9

=> OB = 3 cm

And as Diagonal BD,

BD = OB + OD , where OB = OD = 3 cm

BD = 3 + 3 cm = 6cm.

Therefore the two diagonals of the rhombus are 8 cm and 6cm.

And as we know ,

$\sf{Area \: of \: rhombus \: = \frac{1}{2} \times D_{1}} \times D_{2} \\ \sf{Area \: of \: rhombus \: = \frac{1}{2} \times 8 \times 6} \\ \boxed{\sf{Area \: of \: rhombus \: = 24 \: {cm}^{2}}}$

Therefore:-

The area of rhombus is 24 cm².