The sum of the squares of the measures of the sides of a square is 256. Find the measure of a diagonal of the square.
OH !
they mean all four sides!
256/2 = 128
128 = 64*2 = 8*8*2
sqrt(128) = 8 sqrt 2
sqrt(256) = 16
the answer in the book says 8 sqrt of 2. i don't know how to solve it though
To find the measure of the diagonal of the square, we need to determine the length of the sides of the square first. Let's set the length of one side of the square as 'x'.
According to the problem, the sum of the squares of the measures of the sides of the square is 256. In mathematical terms, this can be represented as:
x^2 + x^2 + x^2 + x^2 = 256
Simplifying the equation:
4x^2 = 256
Dividing both sides of the equation by 4:
x^2 = 64
Taking the square root of both sides:
√x^2 = √64
x = 8
So, the length of each side of the square is 8 units.
Now that we know the length of one side, we can find the length of the diagonal using the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b). In this case, the diagonal represents the hypotenuse, and the two sides of the square represent the other two sides of the triangle.
Using the Pythagorean theorem:
c^2 = a^2 + b^2
In our case, the two sides of the square are both 8 units long. So, substituting the values:
c^2 = 8^2 + 8^2
c^2 = 64 + 64
c^2 = 128
Taking the square root of both sides:
√c^2 = √128
c = √128
Simplifying the square root of 128:
c ≈ 11.314
So, the measure of the diagonal of the square is approximately 11.314 units.