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.)