Geometry: The Pythagorean Theorem
Altitude on Hypotenuse Theorem - If an altitude is drawn to the hypotenuse of a right
triangle, then
1. The two triangles formed are similar to the given right triangle and to each other.
2. The altitude to the hypotenuse is the mean proportional between the segments of the
hypotenuse (x/h=h/y, or h²=xy)
3. Either leg of the given right triangle is the mean proportional between the hypotenuse
of the given right triangle and the segment of the hypotenuse adjacent to that leg (i.e.,
the proportion of that leg on the hypotenuse)
(y/a=a/c, or a²=yc)(x/b=b/c, or b²=xc)
Pythagorean Theorem - The square of the measure of the hypotenuse of a right triangle
is equal to the sum of the squares of the measures of the legs.
a²+b²=c²
Converse of the Pythagorean Theorem - If the square of the measure of one side of a
triangle equals the sum of the squares of the measures of the other two sides, then the
angle opposite the longest side is a right angle.
If c is the length of the longest side of a triangle, and - 1. a²+b²>c², then the triangle is
acute
2. a²+b²=c², then the triangle is right
3. a²+b²<c², then the triangle is obtuse
Distance Formula/Theorem - If P=(x₁,y₁) and Q=(x₂,y₂) are any two point, then the
distance between them can be found with the formula
PQ=√[(x₂-x₁)²+(y₂-y₁)²]
or
PQ=√[(∆x)²+(∆y)²]
Pythagorean Triples - Any three whole numbers that satisfy the equation a²+b²=c² form
a Pythagorean triple.
Ex. (3,4,5) (6,8,10 is in the (3,4,5) family)
(5,12,13)
(7,24,25)
(8,15,17)
(9,40,41)
(11,60,61)
(20,21,29)
(12,35,37)
