Question

the diognal of the rectangal is √41 and area of the rectangal is 20cm square. Find the perimeter of the rectangal.​

Answer 1

Given:

• Length of diagonal = √41 cm
• Area of rectangle =  20 cm

To find:

Perimeter of rectangle = ?

Solution:

By Pythagoras theorem,

(Diagonal)² = (Length)² + (Breadth)²

⇒ (√41)² = l² + b²

⇒ l² + b² = 41     ------- [Equation 1]

We know that;

Area of rectangle = Length × Breadth

→ 20 = lb         ------ [Equation 2]

Now,

(l + b)² = l² + b² + 2lb

➜ (l + b)² = (l² + b²) + 2lb

➟ (l + b)² = 41 + 2(20)

➝ (l + b)² = 41 + 40

➝ (l + b)² = 81

➝ l + b = √81

∴ l + b = 9

For finding Perimeter,

Perimeter = 2(l + b)

➳ Perimeter = 2(9)

∴ Perimeter = 18 cm

Answer 2

$\huge\mathrm{CorrectQuestion ★}$

The diagonal of the rectangle is √41 and area of the rectangle is 20 cm square. Find the perimeter of the rectangle .​

$\huge\mathrm{Answer}$

⚡

Diagonal = √41 cm

Area = 20 cm²

Let the length be - l cm (L)

Let the breadth be - b cm

$\boxed\sf{Pythagoras\: theorem\: ➡ \: Diagonal²\:= \:Length²\:+\:Breadth²}$

[ Since, all the angles of rectangle are 90° , and the length is the base whereas the breadth is the height . So, according to pythagoras theorem, Hypotenuse² = Height² + Base² ]

➡ 41 = l² + b²

➡ b² = 41 - l²

➡ b = √( 41 - l²) _____( i )

★ $\bf\boxed{Area\:= \: Length\: × \:Breadth\:}$

➡ 20 = l × b

➡ b = 20 / l _____ ( ii )

Substituting the value of b from eqn ( i ) to eqn ( ii ) we get,

√(41 - l² ) = 20 / l

➡ 41 - l² = 400 / l² [ Squaring both the side ]

➡ 41l² - l⁴ = 400

➡ l⁴ - 41l² + 400 = 0

➡ l⁴ - ( 25 + 16 ) l² + 400 = 0

➡ l⁴ - 25l² - 16l² + 400 = 0

➡ l² ( l² - 25 ) - 16 ( l² - 25 ) = 0

➡( l² - 16 ) ( l² - 25 ) = 0

l² - 16 = 0

➡ l² = 16

➡l = 4

Also,

l² - 25 = 0

➡ l² = 25

➡ l = 5

So, here both the values are applicable therefore,

Breadth = 20 / length

When l = 5 ,

Breadth = 20 / 5 = 4

When l = 4,

Breadth = 20 / 4 = 5

Therefore,

Perimeter = 5 × 4 = 20 cm ( when l = 4 , b = 5 )

Perimeter = 4 × 5 = 20 cm ( when l = 5 , b = 4 )

As both the results are 20 cm, so the perimeter of the triangle is 20 cm