Sum of the areas of two squares is 400 cm². If the difference of their perimeters is 16 cm, find the sides of the two squares.

Let the side of first square be x and second be y
Area of first square = x²
Area of second square = y²
Perimeter of first square = 4x
Perimeter of second square = 4y
So as per given conditions,
x² + y² = 400 (1)
And 4x - 4y = 16 → 4(x - y) = 16 → x - y = 4 → x = y + 4 (2)
Substituting the value of y in equation (1) we get
(y + 4)² + y² = 400
→ y² + 8y + 16 + y² = 400
→ 2y² + 8y + 16 = 400
→ 2(y² + 4y + 8) = 400
→ y² + 4y + 8 = 200
→ y² + 4y - 192 = 0
This is a quadratic equation. We can solve it by factoring or using the quadratic formula.
The factors of -192 that add up to 4 are 16 and -12. So we can rewrite the equation as:
(y + 16)(y - 12) = 0
So y + 16 = 0 or y - 12 = 0
y = -16 or y = 12
y = -16 is negative, which is not possible as it represents the side of a square.
So y = 12 is the side of the square.
Put value of y = 12 in equation (2) we get
x = 12 + 4 = 16
Hence, sides are 12 cm and 16 cm