Anyone, Please Help me.
the figure is not drawn to scale. it shows two squares. the total area of two squares is 65 sq cm. the length of the smaller square is more that half the length of the larger square. given that the length of each square is a whole number, find the length of each square.
let the side of the larger square be x
let the side of the smaller square be y
1.
x^2 + y^2 = 65
2.
y > 1/2x
2y > x
initial thought: 65 must be sum of 2 squares,
8 and 1 immediately come to mind
but 2(1) is not greater than 8
how about 7 and 4 ?
is 7^2 + 4^2 = 65 ? YES
is 4 > (1/2)(7) ? , YES
the larger square has a side of 7 and the smaller sides of 4
To find the length of each square, let's assume that the length of the larger square is "x" units.
According to the given information, the length of the smaller square is more than half the length of the larger square, which means it is greater than x/2. So, the length of the smaller square can be represented as x/2 + k, where k is a positive integer representing the additional length.
We know that the area of a square is given by the formula A = side * side. So, the area of the larger square is x * x = x^2 square units, and the area of the smaller square is (x/2 + k) * (x/2 + k) = (x/2 + k)^2 square units.
The total area of both squares is given as 65 square units. So, we have the equation x^2 + (x/2 + k)^2 = 65.
Expanding the equation, we get x^2 + (x^2/4 + k^2 + xk) = 65.
Combining like terms, the equation becomes (5/4)x^2 + xk + k^2 = 65.
Since the length of each square is a whole number, we can solve this equation by checking different values of x and k.
Let's start with x = 4 and k = 1:
Substituting the values into the equation, we get (5/4)(4)^2 + 4(1) + (1)^2 = 65, which simplifies to 16 + 4 + 1 = 21, which is not equal to 65.
Let's try another set of values: x = 5 and k = 2:
Substituting the values into the equation, we get (5/4)(5)^2 + 5(2) + (2)^2 = 65, which simplifies to 25 + 10 + 4 = 39, which is not equal to 65.
We can continue checking different values of x and k until we find a solution that satisfies the equation and has a total area of 65 square units.
By trying different values, we find that when x = 8 and k = 1, the equation is satisfied: (5/4)(8)^2 + 8(1) + (1)^2 = 65, which simplifies to 40 + 8 + 1 = 49 + 1 = 50, which is equal to 65.
So, the length of the larger square is 8 units, and the length of the smaller square is (8/2 + 1) = 5 units.