To find the diagonal we need to know the length of one of the square's sides. We know the area and the area of a square is the length of one side squared:

a = s²

so the length of one side of the square is:

s = \( \sqrt{a} \) = \( \sqrt{64} \) = 8

The Pythagorean theorem defines the square of the hypotenuse (diagonal) of a triangle with a right angle as the sum of the squares of the other two sides:

c² = a² + b²
c² = 8² + 8²
c² = 128
c = \( \sqrt{128} \) = \( \sqrt{64 x 2} \) = \( \sqrt{64} \) \( \sqrt{2} \)
c = 8\( \sqrt{2} \)

A circle is a figure in which each point around its perimeter is an equal distance from the center. The radius of a circle is the distance between the center and any point along its perimeter. A chord is a line segment that connects any two points along its perimeter. The diameter of a circle is the length of a chord that passes through the center of the circle and equals twice the circle's radius (2r).

A trapezoid is a quadrilateral with one set of parallel sides.

According to the Pythagorean theorem, the hypotenuse squared is equal to the sum of the two perpendicular sides squared:

c² = a² + b²
c² = 6² + 7²
c² = 36 + 49
c² = 85
c = \( \sqrt{85} \)

The first step to solve a quadratic equation that's set to zero is to factor the quadratic equation:

z² - 25 = 0
(z - 5)(z + 5) = 0

For this expression to be true, the left side of the expression must equal zero. Therefore, either (z - 5) or (z + 5) must equal zero:

If (z - 5) = 0, z must equal 5
If (z + 5) = 0, z must equal -5

So the solution is that z = 5 or -5