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).

The entire length of this line is represented by AD which is AB + BD:

AD = AB + BD

For parallel lines with a transversal, the following relationships apply:

• angles in the same position on different parallel lines equal each other (a° = w°, b° = x°, c° = z°, d° = y°)
• alternate interior angles are equal (a° = z°, b° = y°, c° = w°, d° = x°)
• all acute angles (a° = c° = w° = z°) and all obtuse angles (b° = d° = x° = y°) equal each other
• same-side interior angles are supplementary and add up to 180° (e.g. a° + d° = 180°, d° + c° = 180°)

Applying these relationships starting with b° = 143, the value of x° is 143.

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:

a² + 11a - 31 = 5a - 4
a² + 11a - 31 + 4 = 5a
a² + 11a - 5a - 27 = 0
a² + 6a - 27 = 0

Next, factor the quadratic equation:

a² + 6a - 27 = 0
(a - 3)(a + 9) = 0

For this expression to be true, the left side of the expression must equal zero. Therefore, either (a - 3) or (a + 9) must equal zero:

If (a - 3) = 0, a must equal 3
If (a + 9) = 0, a must equal -9

So the solution is that a = 3 or -9

The Pythagorean theorem defines the relationship between the side lengths of a right triangle. The length of the hypotenuse squared (c²) is equal to the sum of the two perpendicular sides squared (a² + b²): c² = a² + b² or, solved for c, \(c = \sqrt{a + b}\)