GEOMETRY THEOREMS

Geometry theorems are statements that express mathematical relationships and properties concerning shapes, figures, angles, and other geometric elements. These theorems are fundamental in proving various geometric properties.

Ptolemy's theorem:

Ptolemy's theorem is a geometric relationship that applies to cyclic quadrilaterals, a four-sided polygon where all vertices lie on a single circle. The theorem states that the product of the lengths of its diagonals equals the sum of the products of the lengths of its opposite sides.

Diagonals - AC and BD, purple

Line segments - AD and BC, red. CD and AB, blue.

AC x BD = (AD x BC) + (CD x AB)

Example: If AC =7, BD = 9, AD = 3, BC = 11, and CD = 6, what does AB equal?

Solution: Plugging in the values of the given lengths we get (7 x 9) = (3 x 11) + (6 x AB)

AB = 5

Pythagorean's theorem:

Pythagorean Theorem: In a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Mathematically, if a, b, and c are the lengths of the sides of a right triangle, with c being the length of the hypotenuse, then a^2 + b^2 = c^2

Some common pythagorean triples include

(3,4,5) (5, 12, 13) (8, 15, 17) (9, 40, 41)

Example: If a = 11 and b=60, what does c equal?

Solution: (11 x 11) + (60 x 60) = 3721 = 61 x 61