# q3

If diagonal and area of rectangle are 25 m and 168 , then what is the length of the rectangle?
17m
31m
12m
24m

• Angikar Ghosal ·

According to Pythagoras' theorem, in a right angled triangle, hypotenuse = √Base2 + Altitude2

In a rectangle, all angles are 90 degrees. Therefore, the diagonal divides the rectangle into two right triangles.

Therefore Diagonal = √Length2 + Breadth2

Therefore, 25 = √Length2 + Breadth2
Therefore, 252 = (√Length2 + Breadth2)2 [Squaring both Sides]
Therefore, 625 = Length2 + Breadth2
Therefore 2*Length*Breadth = 168*2 = 336
Therefore, Length + Breadth = √961 = 31 [Since Length + Breadth � -31]
Therefore Breadth = 31 - Length

Therefore, Area = Length * Breadth = Length * (31 - Length) = 168

Therefore, -Length2 + 31*Length = 168
Therefore, -Length2 + 31*Length - 168 = 0
Therefore, Length2 - 31*Length + 168 = 0 [Multiplying both sides by (-1)]

Therefore, Length2 - 24*Length -7*Length + 168 = 0

• Angikar Ghosal ·

Continuing the problem,

Therefore, Length(Length - 24) - 7(Length - 24) = 0
Therefore, (Length - 24)(Length - 7) = 0
Since the product of two quantities is 0, either one of them must be zero.

If Length - 24 = 0, Length = 24.
If Length - 7 = 0, Length = 7.
If Length = 24, Breadth = 7.
If Length = 7, Breadth = 24.
But we generally take the length more than the breadth.
Therefore, we rule out the second alternative.
Therefore,

• Angikar Ghosal ·

Extra Notes for the sum,

You can also solve the problem by taking (Length - Breadth)2
as (625-336) or 289 or 172.

In either case, the answer is the same.

You can also solve the quadratic equation by completing the square or by applying the formula.

I have done it for you by factorisation.

• Astha Gupta ·

24m....is the length...

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