Question

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Answer 1

Given

• Length of the rectangle= x
• breath of rectangle= X+1
• diagnoal= 5 cm

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To Find

area of the rectangle

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Formula used

${ \mathtt \color{red}{Hypothenuse ²= Base²+ Leg²}}$

$\mathtt \color{blue}{Area \:of \:Rectangle= length× breath}$

$\mathtt \color{magenta}{(a+b)² =a²+b²+2ab}$

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Concept

~Here the concept of Pythagoras Therom, quadratic equation and property of rectangle is used. We know that each angle in the rectangle is 90° so the diagnoal which divide the rectangle form two right angle triangle. We will apply the Pythagoras Therom and get a quadratic equation through which we will find the value of x. After that we will find the area.

Let's proceed!!

$\mathtt{According\: to \: Pythagoras \:Therom}$

$\sf{ \to \:Hypothenuse²= Base²+ Leg²}$

$\sf{ \implies 5² = x²+(x+1)²}\\ \\ \sf{ \implies25 =x²+x²+1+2x}\\ \\ \sf{ \implies 25=2x²+2x+1}\\ \\ \sf {\implies24=2x²+2x}\\ \\ \sf{\implies 2x²+2x-24=0}\\ \\\sf{\implies {x}^{2} +x- 12 =0 }\\ \\ \sf{\implies \: x²+4x-3x-12=0} \\ \\ \sf{\implies \: x(x+4)-3(x+4) =0} \\ \\ \sf{\implies \: (x-3)(x+4)=0 }$

$\sf{ \implies \: x=3 \: or \: -4}$

x value can never be negative, therefore X = 3

➜Length of rectangle= 3 cm

➜breath of rectangle= 3+1 = 4cm

$\mathtt \color{skyblue}{Area \: of \: Rectangle= length×breath}$

➜Area of Rectangle= 3 ×4

➜Area of Rectangle= 12 cm²

Therefore the area of rectangle is 12cm²

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Answer 2

We know, diagonal of a rectangle = √(l² + b²)

∴ √(l² + b²) = 5

By squaring both sides we get,

⇒ l² + b² = 5²

⇒ (x + 1)² + x² = 25

⇒ x² + 2x + 1 + x² - 25 = 0

⇒ 2x² + 2x - 24 = 0

By dividing (2) from LHS and RHS,

⇒ x² + x - 12 = 0

⇒ x² + (4 - 3)x - 12 = 0

⇒ x² + 4x - 3x - 12 = 0

⇒ x(x + 4) - 3(x + 4) = 0

⇒ (x + 4)(x - 3) = 0

∴ x = - 4 or 3.

As length cannot be negative, thus x = 4 units.

Now we know,

Area of rectangle = l × b

⇒ ar(ABCD) = (x + 1)(x)

⇒ ar(ABCD) = (3 + 1)3 = 12 sq. units.