Question

if the perimeter of a rectangular plot is 68m and length of its diagonal is 26m.find it's area.​

Answer 1

Recall the following formulae to find the side, perimeter and area of a rectangle:

• $\text {Area} = \text {Length} \times \text {Breadth}$

• $\text{Perimeter =} 2(\text{Length} + \text{Breadth})$
• $\text {Pythagoras Theorem : }a^2 + b^2 = c^2$

Given:

Perimeter = 68 m

Diagonal = 26 m

Find the Breadth in term of the Length:

$\text{Perimeter =} 2(\text{Length} + \text{Breadth})$

$2(\text{Length} + \text{Breadth}) = 68$

$\text{Length} + \text{Breadth} = 34$

$\text{Breadth} = 34 - \text{Length}$

Find the Length:

$a^2 + b^2 = c^2$

$L^2 + (34 - L)^2 = 26^2$

$L^2 + 34^2 + L^2 - 68L = 26^2$

$2L^2 - 68L + 34^2 - 26^2 = 0$

$2L^2 - 68L + 480 = 0$

$2(L - 10)(L - 24) = 0$

$L = 10 \text{ or } L = 24$

Find the breadth:

$\text{length} = 24$

$\text{breadth} = 34 - 24$

$\text{breadth} = 10$

Find the area:

$\text {area} = \text {length} \times \text {breadth}$

$\text {area} = 24 \times 10$

$\text {area} = 240 \text { m}^2$

Answer: The area is 240 m².

Answer 2

The Area of the rectangular plot is 240 sq meters .

Step-by-step explanation:

Given as :

The perimeter of a rectangular plot = 68 meters

The Length of its diagonal = d = 26 meters

Let The Length of the rectangle = L meters

Let The breadth of the rectangle = B meters

Let The Area of the rectangle = A sq meters

According to question

From figure  , ABCD is a rectangle  , BD is its diagonal

∵   Perimeter of a rectangular plot = 2 ( Length + Breadth )

or,                                                68 = 2 ( L + B )

Or,                                            L + B = $\dfrac{68}{2}$

∴                                               L + B = 34 meters            ......1

Again

As The Rectangle is formed by adding two triangles i.e Δ ABD + Δ CBD

From Triangle ABD

From Pythagoras theorem

Hypotenuse² = Perpendicular² + Base²

BD² = AD² + AB²

i.e  d² = B² + L²

Or,  B² + L² = 26²

or,   ( L + B ) ² - 2 L B = 26²

Or, 2 L B = 34² - 26²                                   (from eq 1 )

or,   2 L B = 1156 - 676

or,   2 L B = 480

or, L B = $\dfrac{480}{2}$

∴  L B = 240               ...........2

From eq 1 and eq 2

(L - B)²  = (L + B)² - 4 L B

or, (L - B)²  = (34)² - 4 × 240

or, (L - B)²  = 1156 - 960

or, (L - B)²  = 196

∴   L - B = √196

i.e  L - B = 14       meters            ..........3

Again from eq 2 and eq 3

( L + B ) + ( L - B ) = 34 + 14

Or,  2 L = 48

∴        L = $\dfrac{48}{2}$

i,e Length = L = 24 meters

And, put the value of L in eq 3 , we get

i.e   24 - B = 14       meters    

Or,  B = 24 - 14

So , Breadth = B = 10 meters

Again

∵ Area of Rectangular plot = Length × Breadth

i.e  A = 24 meters × 10 meters

∴   Area = 240 sq meters

So, The Area of the rectangular plot = A = 240 sq meters

Hence,  The Area of the rectangular plot is 240 sq meters . Answer