Angles around a line add up to 180°. Angles around a point add up to 360°. When two lines intersect, adjacent angles are supplementary (they add up to 180°) and angles across from either other are vertical (they're equal).

An acute angle measures less than 90°, a right angle measures 90°, and an obtuse angle measures more than 90°.

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

AD = AB + BD

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 = \( \sqrt{a} \) = \( \sqrt{81} \) = 9

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 = 9² + 9²
c² = 162
c = \( \sqrt{162} \) = \( \sqrt{81 x 2} \) = \( \sqrt{81} \) \( \sqrt{2} \)
c = 9\( \sqrt{2} \)

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

y² - 15y + 56 = 0
(y - 7)(y - 8) = 0

For this expression to be true, the left side of the expression must equal zero. Therefore, either (y - 7) or (y - 8) must equal zero:

If (y - 7) = 0, y must equal 7
If (y - 8) = 0, y must equal 8

So the solution is that y = 7 or 8