Angles and Polygons

The Midpoint Formula

The Midpoint Formula

The midpoint “M” of a line segment with

endpoints P(x1, y1) and Q(x2, y2) has

coordinateskjlkjlkjlkjlkjlkjlkjlkj 1 2 1 2,2 2

x x y y

The Midpoint Formula

Example 1: Find the coordinates of the midpoint of the line segment joining (-8, 3) and (5, 11).

3,7

2

Solution: The midpoint is:

8 5 3 11,

2 2

1 2 1 2,2 2

x x y y

The Distance Formula

The Distance Formula

The distance, “d”, between points P(x1, y1)

and Q(x2, y2) is given by the formula

2 21 2 1 2( ) ( )d x x y y

The Distance Formula

Example 1: Find the distance between thepoints (-3, -2) and (6, -5). Express youranswer in simplest radical form.Solution:

2 2( 9) (3)d

2 2( 3 6) ( 2 ( 5))d

81 9d

90d

3 10d

2 21 2 1 2( ) ( )d x x y y

Angles

The stationary ray is called the initial side (arm) of the angle

The revolving ray is called the terminal side (arm)

The fixed point is the vertex

initial side

terminal side

vertex

Angles

If the direction of rotation is counterclockwise the angle is said to be a positive angle

If the direction of rotation is clockwise, the angle is said to be a negative angle

A positive angle A negative angle

Angles

An angle has been drawn in standard

position when its vertex is located at the

origin and when the initial side of the

angle coincides with the positive x-axis.

x

y

Standard Position

Angles

Coterminal Angles Formula

If θ is an angle drawn in standard position,

then, θ + n(360°) is coterminal with, where

“n” is any integer.

Angles

Example: Determine other angles which are coterminal to 155°.

Solution: θ = 155°

θ + n(360°) = 155° + 1(360°) = 515°

= 155° + 2(360°) = 875°

= 155° + (-1)(360°) = -205°

Homework

Do #1 – 11 on page 222 from Section 7.1 and do # 1 – 17 odd #’s only, 21, 23, 25, and 37 on pages 227 and 228 from Section 7.2 for Friday June 5th