q3

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

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
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

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 (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.