Geometry Chapter 8

Geometric Mean

The positive square root of the product of two numbers

Theorem 8-1-1

The altitude to the hypotenuse of a right triangle forms two triangles that are similar to each other and to the original triangle.

EX: ∆PIE~∆ISE~∆PSI

Corollary 8-1-2

The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the two segments of the hypotenuse.

EX: y/h=h/x

Corollary 8-1-3

The length of a leg of a right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse adjacent to that leg.

EX: c/a=a/y and c/b=b/x

Trigonometric Ratio

Ratio of two sides of a right triangle

Sine

The ratio of the length of the leg opposite the angle to the length of the hypotenuse

Cosine

The ratio of the length of the leg adjacent the angle to the length of the hypotenuse

Tangent

The ratio of the length of the leg opposite the angle to the length of the leg adjacent to the angle

Angle of Elevation

Angle formed by a horizontal line and observer’s line of sight to an object above the horizontal line

Angle of Depression

Angle formed by a horizontal line and an observer’s line of sight to an object below the horizontal line

Law of Sines

sinA/a=sinB=b=sinC=c

Law of Cosines

c²=a²+b²-2ab(cosC)

Law of Sines

Use ____ when you are given two angles and one unincluded side, or two sides and one unincluded angle

Law of Cosines

Use ____ when given all three sides lengths, or two sides and one unincluded angle

Vector

A quantity that has both length and direction

Initial Point

Where a vector starts

Terminal Point

Where a vector ends

Component Form

<x,y>

Magnitude

The length of a vector, written as |→|

Direction

The angle formed with the vector and the x-axis

Numerical Vector Addition

<x₁+x₂, y₁+y₂>