Chapter 8 Similar Right Triangles

Theorem 9.4 Pythagorean theorem

In a right triangle, the square of the length of the hypotenuse is equal to the sum of the square of the lengths of the legs. c^2 = a^2 + b^2

Right triangle, Obtuse, Acute

Right triangle- c^2 = a^2 + b^2 Obtuse triangle- c^2 > a^2 + b^2 Acute triangle- c^2 < a^2 + b^2

SOH CAH TOA

SOH (Sin) - Sin Opposite/Hypotenuse CAH (Cos)- Cos Adjacent/Hypotenuse TOA (Tan)- Tan Opposite/Adjacent

Theorem 9.1

If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other

Theorem 9.5 Converse of the pythagorean theorem

If the square of the length of the longest side of a triangle is EQUAL to the sum of the squares of the lengths of the other two sides, then the triangle is a RIGHT triangle. If c^2 = a^2 + b^2, then triangle ABC is a right triangle

Theorem 9.7

If the square of the length of the longest side of a triangle is GREATER than the sum of the squares of the lengths of the other two sides, then the triangle is OBTUSE. If c^2 > a^2 + b^2, then triangle ABC is obtuse

Theorem 9.6

If the square of the length of the longest side of a triangle is LESS than the sum of the squares of the lengths of the lengths of the other two sides, then the triangle is ACUTE. If c^2 < a^2 + b^2, then triangle ABC is acute

Theorem 9.9: 30-60-90 triangle

In a 30˚- 60˚- 90˚ triangle, the hypotenuse is twice as long as the shortest leg, and the longer leg is √ times as long as the shorter leg Hypotenuse = 2 x shortest leg Longer leg = √3 x shorter leg

Theorem 9.8: 45-45-90 triangle

In a 45˚- 45˚- 90˚triangle, the hypotenuse is √2 times as long as each leg. Hypotenuse = √2 x leg

Theorem 9.3

In a right triangle, the altittude from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of each leg of the right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg

Theorem 9.2

In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of the altitude is the geometric mean of the lengths of the two segments

Special right triangles

A special right triangle is a right triangle with some regular feature that makes calculations on the triangle easier, or for which simple formulas exist. For example, a right triangle may have angles that form a simple ratio, such as 45-45-90.

Trigonometric ratio

a ratio of the lengths of two sides of a right triangle

Pythagorean triple

a set of three positive integers (whole numbers) a, b, c which form the sides of a right angled triangle and fit the rule a^2 + b^2 = c^2

Angle of elevation

angle that your line of sight makes with a horizontal line when you stand and look up at a point

Cosine

(Cos) a trigonometric ratio- adjacent/hypotenuse

Sine

(Sin) a trigonometric ratio- opposite/hypotenuse

Tangent

(Tan) a trigonometric ratio- opposite/adjacent