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

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4 Answers
According to Pythagoras' theorem, in a right angled triangle, hypotenuse = √Base^{2} + Altitude^{2}
In a rectangle, all angles are 90 degrees. Therefore, the diagonal divides the rectangle into two right triangles.
Therefore Diagonal = √Length^{2} + Breadth^{2}
Therefore, 25 = √Length^{2} + Breadth^{2}
Therefore, 25^{2} = (√Length^{2} + Breadth^{2})^{2} [Squaring both Sides]
Therefore, 625 = Length^{2} + Breadth^{2}
Area = Length*Breadth
168 = Length*Breadth
Therefore 2*Length*Breadth = 168*2 = 336
Therefore, Length^{2} + 2*Length*Breadth + Breadth^{2} = 625 + 336
Therefore, (Length + Breadth)^{2}= 961
Therefore, Length + Breadth = √961 = 31 [Since Length + Breadth â‰ 31]
Therefore Breadth = 31  Length
Therefore, Area = Length * Breadth = Length * (31  Length) = 168
Therefore, Length^{2} + 31*Length = 168
Therefore, Length^{2} + 31*Length  168 = 0
Therefore, Length^{2}  31*Length + 168 = 0 [Multiplying both sides by (1)]
Therefore, Length^{2}  24*Length 7*Length + 168 = 0
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,
Answer) Length = 24.
Extra Notes for the sum,
You can also solve the problem by taking (Length  Breadth)^{2}
as (625336) or 289 or 17^{2}.
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.