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A 34 cm long wire is bent into a rectangle. The length of its diagonal is 13 cm. What are the lengths of the sides of the rectangle?

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14 cm and 3 cm

16 cm and 7 cm

12 cm and 5 cm

15 cm and 2 cm

answer is C.

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Detailed Solution

Given,
Wire length = 34 cm
Diagonal length = 13 cm
Now perimeter of rectangle is equal to wire length. Thus, perimeter of rectangle

P = Wire Length

2 (Length + Breath) = 34 cm

Length + Breath = 17 cm

Let length of rectangle = x cm.
Thus, breadth of rectangle
= (17 - x) cm
F:New DownloadQ.90.jpg

As diagonal is 13 cm. Thus, from Pythagoras theorem,

x^{2} + (17 - x)^{2} = 13^{2}

x^{2} + 289 - 34 x + x^{2} = 1692

x^{2} - 34 x + 120 = 0

x^{2} - 17 x + 60 = 0

Value of discriminant,

b^{2} - 4 ac = 289 - 240

= 49

Use the quadratic formula as,

x = \frac{- b \pm \sqrt{b^{2} - 4 ac}}{2 a}

x = \frac{17 \pm \sqrt{49}}{2}

x = \frac{17 \pm 7}{2}

Thus,

x = 12 or 5

At length, x = 12 cm,
Breadth, 17 - x = 5 cm
At length, x = 5 cm,
Breadth, 17 - x = 12 cm
So, the lengths of sides of rectangle are 12 cm and 5 cm.
Correct option is 3.

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