26924m....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
69According 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
Area = Length*Breadth
168 = Length*Breadth
Therefore 2*Length*Breadth = 168*2 = 336
Therefore, Length2 + 2*Length*Breadth + Breadth2 = 625 + 336
Therefore, (Length + Breadth)2= 961
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
69Continuing 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.
69Extra 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.
