A rhombus is a parallelogram with four equal-length sides. A square is a rectangle with four equal-length sides.

The first step to solve a quadratic expression that's not set to zero is to solve the equation so that it is set to zero:

z² - 10z + 23 = -z + 3
z² - 10z + 23 - 3 = -z
z² - 10z + z + 20 = 0
z² - 9z + 20 = 0

Next, factor the quadratic equation:

z² - 9z + 20 = 0
(z - 4)(z - 5) = 0

For this expression to be true, the left side of the expression must equal zero. Therefore, either (z - 4) or (z - 5) must equal zero:

If (z - 4) = 0, z must equal 4
If (z - 5) = 0, z must equal 5

So the solution is that z = 4 or 5

The formula for area is πr²:

a = πr²
a = π(3²)
a = 9π

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\( \sqrt{2} \)

An exterior angle of a triangle is equal to the sum of the two interior angles that are opposite:

d° = b° + c°

To find angle c, remember that the sum of the interior angles of a triangle is 180°:

180° = a° + b° + c°
c° = 180° - a° - b°
c° = 180° - 29° - 61° = 90°

So, d° = 61° + 90° = 151°

A shortcut to get this answer is to remember that angles around a line add up to 180°:

a° + d° = 180°
d° = 180° - a°
d° = 180° - 29° = 151°