1. Y. Davis Geometry Notes
Chapter 8

2. Geometric mean
The positive square root of the product of two numbers.

3. Theorem 8.1
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.

4. Theorem 8.2 Geometric Mean (Altitude) Theorem
The altitude drawn to the hypotenuse of a right triangle separates the hypotenuse into two segments. The length of this altitude is the geometric mean between the lengths of these two segments.

5. Theorem 8.3 Geometric Mean (Leg) Theorem
The altitude drawn to the hypotenuse of a right triangle separates the hypotenuse into 2 segments. The length of a leg of this triangle is the geometric mean between the length of the hypotenuse and the segment of the hypotenuse adjacent to that leg.

7. Theorem 8.4 Pythagorean Theorem
In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.
a²+b²=c²

8. Pythagorean Triple
3 whole numbers that satisfy the Pythagorean Theorem.

9. Theorem 8.5 Converse of the Pythagorean Theorem
If the sum of the squares of the lengths of the shortest sides of a triangle is equal to the square of the length of the longest side, then the triangle is a right triangle.
a²+b²=c²

10. Theorem 8.6 Acute Triangle Theorem
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 other two sides, then the triangle is an acute triangle.
a²+b²>c²

11. Theorem 8.7 Obtuse Triangle Theorem
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, hen the triangle is an obtuse triangle.
a²+b²<c²

12. Theorem 8.8— Triangles
In a triangle, the legs l are congruent and the length of the hypotenuse h is times the length of a leg.

13. Theorem 8.9— Triangle
In a triangle, the length of the hypotenuse h is 2 times the length of the shorter leg s, and the length of the longer leg l is times the length of the shorter leg.

14. Trigonometry
The study of the measures of triangles.

15. Trigonometric Ratios
Sine (sin)
Cosine (cos)
Tangent (tan)

16. Inverse Trigonometric Ratios

17. Angle of Elevation
The angle formed by a horizontal line and an observer’s line of sight to an object above the horizontal line.

18. Angle of Depression
The angle formed by a horizontal line and an observer’s line of sight to an object below the horizontal line.

19. Law of Sines
Can be used to find side lengths and angle measures for any triangle.

20. Theorem 8.10—Law of Sines
If , with lengths of a, b, and c, representing the lengths of the sides opposite the angles with measure A, B,and C, then

21. Theorem 8.11—Law of Cosines
If , with lengths of a, b, and c, representing the lengths of the sides opposite the angles with measure A, B,and C, then

22. Vector
A quantity that has both magnitude and direction.

23. Magnitude of a vector Size
Length from initial point to terminal point.

24. Initial Point
Starting point of the vector

25. Terminal point
Ending point of a vector

26. Direction of a vector The angle it forms with the horizontal line. Or
A measurement between 0 and 90 degrees east or west of the north-south line.

27. Resultant
A single vector that is the sum of 2 or more vectors

28. Parallel vectors
Have the same or opposite directions, but not necessarily the same magnitude

29. Opposite vectors
Have the same magnitude, but opposite directions.

30. Equivalent vectors
Have the same magnitude and direction.

31. Standard position
When the initial point of a vector is the origin.

32. Component form

33. Vector Operations