Find the length of the diagonal and area of a Rectangle whose sides are 10 cm and 12 cm. If the perimeter of a square is 2 ( 2x+4y) , find the area ?

The correct answer is: Length of the diagonal and area of the given Rectangle is 2 √61 cm & 120 cm2, respectively. Area of the given square is (x + 2y)2 cm2.

step-by-step solution:-
In the adjacent diagram, we can see that the diagonal divides the given rectangle into 2 right angled triangles.
Also, diagonal of the rectangle becomes the hypotenuse of the 2 triangles.
Hypotenuse of the given triangles = diagonal of the given rectangle
Now, sides of the rectangle = sides of the triangle (other than hypotenuse)
By applying Pythagorean theorem, For a right angled triangle-
Hypotenuse2 = sum of the squares of the remaining 2 sides
∴ Hypotenuse2 = 122 + 102
∴ Hypotenuse2 = 144 + 100
∴ Hypotenuse2 = 244
∴ Hypotenuse = 2 √61 cm .............................. (Taking square root both the sides)
Area of the given rectangle = length × breadth
∴ Area of the given rectangle = 12 × 10 ...................................................................................... (From given information & Equation i)
∴ Area of the given plot = 120 cm2
Perimeter of the given square = 4 × sides
∴ 2 (2x + 4y) = 4 × sides
∴ 4x + 8y = 4 × sides
∴ x + 2y = side .................. (Dividing both sides by 4) ................................ (Equation i)
Now, Area of the given square = side2
∴ Area of the given square = (x + 2y)2 ......................................................................................... (From Equation i)
Final Answer:-
∴ Length of the diagonal and area of the given Rectangle is 2 √61 cm & 120 cm2, respectively.
Area of the given square is (x + 2y)2 cm2.