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 line on the coordinate grid can be defined by a slope-intercept equation: y = mx + b. For a given value of x, the value of y can be determined given the slope (m) and y-intercept (b) of the line. The slope of a line is change in y over change in x, \({\Delta y \over \Delta x}\), and the y-intercept is the y-coordinate where the line crosses the vertical y-axis.

To factor a quadratic expression, apply the FOIL method (First, Outside, Inside, Last) in reverse. First, find the two Last terms that will multiply to produce -24 as well and sum (Inside, Outside) to equal 5. For this problem, those two numbers are -3 and 8. Then, plug these into a set of binomials using the square root of the First variable (y²):

y² + 5y - 24
y² + (-3 + 8)y + (-3 x 8)
(y - 3)(y + 8)

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).

You need to find the value of a so solve the first equation in terms of z:

-7a + z = -1
z = -1 + 7a

then substitute the result (-1 - -7a) into the second equation:

7a + 9(-1 + 7a) = -6
7a + (9 x -1) + (9 x 7a) = -6
7a - 9 + 63a = -6
7a + 63a = -6 + 9
70a = 3
a = \( \frac{3}{70} \)
a = \(\frac{3}{70}\)