Question

if a rhombus has a perimeter of 60cm and one of its diagonal measures 20 cm then find the area of rhombus using herons formula​

Answer 1

Sᴏʟᴜᴛɪᴏɴ :-

→ Perimeter of Rhombus = 60cm.

→ So, Each side of Rhombus = (Perimeter/4) = (60/4) = 15cm.

And, Now, we know That, Any Diagonal of a Rhombus Divide it into Two Equal Isosceles ∆ area.

So, Now, we Have :-

→ An Isosceles ∆ with sides , 15cm,15cm & 20cm.

So,

→ Semi-Perimeter of Isosceles ∆ = s = (15 + 15 + 20)/2 = 25cm.

Therefore,

→ Area of Isosceles ∆ with Heron's formula = √[s(s - a)(s - b)(s - c)]

Putting values Now, we get :-

→ Area of Isosceles ∆ = √[25*(25-15)*(25-15)*(25-20)]

→ Area of Isosceles ∆ = √[25 * 10 * 10 * 5]

→ Area of Isosceles ∆ = 10√(5 * 5 * 5)

→ Area of Isosceles ∆ = 10 * 5 * √5

→ Area of Isosceles ∆ = 50√5 cm².

Hence,

→ Area of Rhombus = 2 * Area of Isosceles ∆ = 50√5 * 2 = 100√5 cm². (Ans.)

Answer 2

$\maltese$ ${\red{\bold{\underline{SOLUTION : }}}}$ $\maltese$

•The Perimeter of Rhombus = 60cm.

•Each side of Rhombus = (Perimeter/4) = (60/4) = 15cm.

=> As we know that, Any Diagonal of a Rhombus Divide it into Two Equal Isosceles ∆ area.

=> .°. First & Second side = 15cm , and third side = 20cm.

Now,

=> An Isosceles ∆ with sides , 15cm,15cm & 20cm.

•Now, find semi perimeter of isocleles Δ .

=> .°. Semi-Perimeter of Isosceles ∆ = s = (15 + 15 + 20)/2 = 25cm.

=> .°. S = 25cm.

Now, we have to find out Area of Isosceles ∆ with Heron's formula.

=> We know that Heron's formula = √[s(s - a)(s - b)(s - c)]

[Putting the values ] .

=>.°. Area of Isosceles ∆ = √[25×(25-15)×(25-15)×(25-20)]

=> Area of Isosceles ∆ = √[25 ×10 × 10 ×5]

=> Area of Isosceles ∆ = 10√(5 ×5 ×5)

=>Area of Isosceles ∆ = 10 × 5 ×√5

=> Area of Isosceles ∆ = 50√5 cm².

Therefore,

=> Area of Rhombus = 2 ×Area of Isosceles Δ

=> Area of Rhombus = 2 × 50√5 cm²

=> Area of Rhombus = 100√5 cm².

Hence, The Area of rhombus is 100√5 cm² .