Right Triangles
Obtuse Triangle Theorem
If the sum of the squares of the length of the two shorter sides of a triangle is smaller than the square of the length of the largest side, then the triangle is obtuse. (Side 1)^2 + (Side 2)^2 < (Longest Side)^2
Acute Triangle Theorem
If the sum of the squares of the length of the two smaller size of a triangle is larger than the square of the length of the longest side, then the triangle is acute. (Side 1)^2 + (Side 2)^2 > (Longest Side)^2
30°-60°-90° Triangle Theorem
If x is the length of the side opposite the 30° angle, the side opposite the 60° angle equals x square root of 3 (approximately 2.73) and the side opposite the 90° angle equals 2x.
Right Triangle Leg Theorem
In a right angle, when you draw the altitude from the right angle to the hypotenuse, the length of each leg is the geometric mean between the entire hypotenuse and the part of that hypotenuse nearest the leg.
Altitude Length Theorem
In a right triangle, the length of the altitude from the right angle to the hypotenuse is the geometric mean between the two segments of the hypotenuse. (1st Part Of Hypotenuse/Altitude Of Hypotenuse = Altitude Of Hypotenuse/2nd Part Of Hypotenuse)
Pythagorean Theorem
In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs. If the triangle is right then (Leg 1)^2 + (Leg 2)^2 = (Hypo)^2
Converse Of The Pythagorean Theorem
In a triangle, if the sum of the squares of the two smaller sides is equal to the square of the largest side, the triangle is right. If (Side 1)^2 + (Side 2)^2 = (Longest Side)^2 then the triangle is right.
Left Leg Formula
Left Part Of Hypo/Left Leg = Left Leg/Whole Hypo
Cosine (cos)
Length Of Adjacent Leg/Length Of Hypotenuse
Tangent (tan)
Length Of Opposite Leg/Length Of Adjacent Leg
Sine (sin)
Length Of Opposite Leg/Length Of Hypotenuse
Adjacent Side
Next to the angle.
Right Leg Formula
Right Part Of Hypo/Right Leg = Right Leg/Whole Hypo
Hypotenuse
Side across from the right angle.
Opposite Side
Side opposite of the angle.
45°-45°-90° Triangle Theorem
The length of the hypotenuse equals the length of a leg multiplied by square root of 2 (approximately 1.4).
Geometric Mean
The number which lies between two positive numbers.
Law Of Cosines
The square of one side of a triangle is related to the other two sides and the included angle of the triangle. a^2 = b^2 + c^2 - 2ab(cos A), b^2 = a^2 + c^2 - 2ac(cos B), c^2 = a^2 + b^2 - 2ab(cos C)
Altitude Theorem
When the altitude is drawn from the right angle to the hypotenuse of a right triangle, the two triangles that are formed are similar to the original triangle and similar to each other.
Geometric Mean Formula
1 Part Of Hypo/Altitude Of Hypo = Altitude Of Hypo/Other Part Of Hypo
Law Of Sines
sin A/a = sin B/b = sin C/c (Each ratio formed by diving the sine of an angle by the length of the side opposite that angle.)