Chapter 8

• Geometric mean


  If a, b, and x are positive integers and a/x = x/b, then x is called the geometric mean between a and b
• 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.
• Corollary 1

  When the altitude is drawn to the hypotenuse of a right triangle, the length of the altitude is the geometric mea between the segments of the hypotenuse
• Corollary 2

  When the altitude is drawn to the hypotenuse of a right triangle, then each leg is the geometric mean between the hypotenuse and the segment of the hypotenuse adjacent to that leg
• Pythagorean Theorem

  In a right triangle, the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse
• Pythagorean Triples

  1. 3-4-5
  2. 5-12-13
  3. 8-15-17
  4. 7-24-25
  5. 9-40-41
  6. 20-21-29
• If cˆ2 > aˆ2 + bˆ2, then m<C > 90 and triangle ABC is obtuse.
• If cˆ2 < aˆ2 + bˆ2, then m<C < 90 and triangle ABC is acute.
• If cˆ2 = aˆ2 + bˆ2, then m<C = 90 and triangle ABC is a right triangle.
• 45-45-90 triangle


  Triangle where the length of the hypotenuse is √2 times the length of a leg.

  legs = x

  hypotenuse = x√2
• 30-60-90 triangle


  Triangle where the length of the hypotenuse is 2 times the length of the shortest leg and the length of the longer leg is √3 times the length of the shortest leg.

  (30) shorter leg = x

  (60) longer leg = x√3

  (90) hypotenuse = 2x
• Sin

  SOH; opposite / hypotenuse
• Cos

  CAH; adjacent / hypotenuse
• Tan

  TOA; opposite over adjacent
• tan 45˚

  1
• cos 45˚

  √2/2
• sin 45˚

  √2/2
• tan 60˚

  √3
• tan 30˚

  √3/3
• cos 60˚

  1/2
• cos 30˚

  √3/2
• sin 60˚

  √3/2
• sin 30˚


  1/2