Answer: 5 cm and 9 cm
Step-by-step explanation:
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 and 81
x = 5 and 9
yep, 5*9 = 45
Answer:
The lengths of sides of squares are 5 cm and 9 cm.
Step-by-step explanation:
Let the sides of rectangle be x and y .
Area of rectangle = Length × Breadth
Given : Area of rectangle = 45 cm².
⇒ x × y = 45
⇒ [tex]x=\frac{45}{y}[/tex] ........(1)
Two squares are constructed such that two adjacent sides of the rectangle
so squares have side x and y .
Area of square = side × side
Area of square with side x = x²
Area of square with side y = y²
Also, The combined area of the two squares is 106 cm².
⇒ x² + y² = 106
From (1) put value of x , we get,
[tex](\frac{45}{y})^2+y^2=106[/tex]
Solving for y ,
[tex](\frac{45}{y})^2+y^2=106[/tex]
[tex]2025+y^4=106y^2[/tex]
[tex]y^4-106y^2+2025=0[/tex]
[tex]y^4-81y^2-25y^2+2025=0[/tex]
[tex]y^2(y^2-81)-25(y^2-81)=0[/tex]
[tex](y^2-25)(y^2-81)=0[/tex]
[tex]y^2-81=0[/tex] or [tex]y^2-25=0[/tex]
on solving we get y = 5 and y = 9
Also, x can be find by putting in (1),
⇒ [tex]x=\frac{45}{5}=9[/tex] and ⇒ [tex]x=\frac{45}{9}=5[/tex]
⇒ x = 9 and x = 5
Thus, lengths of sides of squares are 5 cm and 9 cm.
