The area of a rectangle is 45 cm2. Two squares are constructed such that two adjacent sides of the rectangle are each also the side of one of the squares. The combined area of the two squares is 106 cm². Find the lengths of the sides of the squares.
If the two squares are x and y, then
xy = 45
x^2 + y^2 = 106
so,
x^2 + (45/x)^2 = 106
x^2 + 2025/x^2 = 106
x^4 - 106x^2 + 2025 = 0
(x^2-25)(x^2-81) = 0
x^2 = 25 or 81
x = 5 or 9
yep, 5*9 = 45
To find the lengths of the sides of the squares, we can start by finding the dimensions of the rectangle.
Let's assume the length of the rectangle is L cm and the breadth is B cm.
We are given that the area of the rectangle is 45 cm², so we can write the equation:
Length × Breadth = Area
L × B = 45
Now, we are given that two adjacent sides of the rectangle are each also the side of one of the squares. Let's call the side of one of the squares S cm.
So, the length of the rectangle can be expressed as L = S + B.
Now we have two equations:
L × B = 45 ------ Equation 1
L = S + B ------ Equation 2
To solve these equations, we need to eliminate the variables L and B.
From Equation 2, L = S + B, we can substitute this value of L into Equation 1:
(S + B) × B = 45
Expanding the equation, we get:
S × B + B² = 45
Now, we know that the combined area of the two squares is 106 cm². Since two adjacent sides of the rectangle are each also the side of one of the squares, the area of one square can be expressed as S².
So, the combined area of the two squares is:
S² + S² = 106
2S² = 106
Now we can solve for S.
Dividing both sides by 2, we get:
S² = 53
Taking the square root of both sides, we get:
S = √53
So, the length of the sides of the squares is √53 cm.