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