Question

The length of a rectangle is 3cm more than thrice its breath . its diagonal is 1cm more than the length of a rectangle. what are the length and breath of the rectangle? (Hint: let the breath=x and length=3x+3)​

Answer 1

GivEn:

• Length of rectangle is 3 cm more than thrice the breadth
• Diagonal of Rectangle is 1 cm more than the length.

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To find:

• Length and breadth of rectangle?

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☯‎‎‎‎‎‎ ‎‎‎‎‎‎Let Breadth of rectangle be x cm.

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Therefore,

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Length of Rectangle is,

3 cm more than thrice the breadth

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$:\implies\sf 3 \times x + 3$

$:\implies\sf \red{3x + 3}$

Also, Diagonal of Rectangle is 1 cm more than length,

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$:\implies\sf (3x + 3) + 1$

$:\implies\sf \red{3x + 4}$

We know that each angle of a rectangle is of 90°.

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Therefore,

★ In right angled ∆ ABC,

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• AB = Perpendicular = x cm
• BC = Base = 3x + 3 cm
• AC = Hypotenuse = 3x + 4 cm

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$\setlength{\unitlength}{1.5cm}\begin{picture}(8,2)\thicklines\put(7.7,3){\tt\large{D}}\put(7.7,1){ \tt\large{C}}\put(9.3,0.7){\sf{\large{3x + 3 cm}}}\put(11.5,1){ \tt\large{B}}\put(8,1){\line(1,0){3.5}}\put(8,1){\line(0,2){2}}\put(11.5,1){\line(0,3){2}}\put(8,3){\line(3,0){3.5}}\put(8.4,2){\sf{\large{3x + 4 cm}}}\qbezier(8,1)(8,1)(11.5,3)\put(11.5,3){ \tt\large{A}}\put(11.3,1){\line(0,2){0.2}}\put(11.3,1.2){\line(2,0){0.2}}\put(11.8,2){\sf{\large{x cm}}}\end{picture}$

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${\underline{\sf{\bigstar\;Using\; Pythagoras\; Theorem\;:}}}$

$\star\;{\boxed{\sf{\purple{H^2 = B^2 + P^2}}}}$

$:\implies\sf (3x + 4)^2 = (3x + 3)^2 + x^2$

$:\implies\sf (3x)^2 + (4)^2 + 2 \times 3x \times 4 = (3x)^2 + (3)^2 + 2 \times 3x \times 3 + x^2$

$:\implies\sf \cancel{9x^2} + 16 + 24x = \cancel{9x^2} + 9 + 18x + x^2$

$:\implies\sf x^2 + 18x + 9 - 24x - 16 = 0$

$:\implies\sf x^2 - 6x - 7 = 0$

Now, Splitting middle term,

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$:\implies\sf x^2 + x - 7x - 7 = 0$

$:\implies\sf x(x + 1) - 7(x + 1) = 0$

$:\implies\sf (x + 1)(x - 7) = 0$

$:\implies\sf x = - 1\;or\; x = 7$

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We know that,

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Dimensions of Rectangle can't be negative.

So, Breadth of Rectangle = 7 cm

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Therefore,

• Breadth, x = 7 cm
• Length, (3x + 3) = 3 × 7 + 3 = 21 + 3 = 24 cm
• Diagonal, (3x + 4) = 3 × 7 + 4 = 21 + 4 = 25 cm