Geometry

Chapter 13 Review

1 1( , )x y 2 2( , )x y

2 22 1 2 1( ) ( )d x x y y

The distance d between points and is:

Example 2Find the distance between (–3, 4) and (1, –4).

Why? Let’s try an example to find out!

22 )4(413

6416

8054

(-3, 4).

. (1, -4)

4

8

Pythagorean Theorem!

4√5

2 2 2( ) ( )x a y b r An equation of the circle with center (a, b) and radius r is:

Let’s analyze (x – 0)2 + (y – 0)2 = 81 to see if it really is a circle!!

How could this be a circle?

2 2( 2) ( 4) 9x y

Find the center and radius of each circle. Sketch the graph.

4. 5.

Center: (2, -4)Radius = 3

.

Example 1b: Find the slope of the line.Example 1b: Find the slope of the line.

-5 – (-2)=

3 – (- 1) x

y

.

.(-1 , -2)

(3 , -5)

y2 – y1=

x2 – x1

slope

- 3=

4

The slope of the line is3

4- __

Positive SlopeGreater than 1

Uphill

Steep

Positive SlopeLess than 1

Uphill

Flatter

Negative SlopeGreater than 1

Downhill

Steep

Negative SlopeLess than 1

Downhill

Flatter

Slope = 0

Undefined Slope

Running up the hill is undefined!

A line with slope 4/3 passes through points (4, -5) and (-2, __ ).

Use the slope formula to find the missing y coordinate.

43

=y – (-5)

-2 – 4

43

=y + 5

-6

Simplify and solve as a proportion

-24 = 3y + 15

-39 = 3yy = -13

-13 y

• Parallel lines have slopes that are equal.

• Perpendicular lines have slopes that are opposite inverses(change the sign and flip).

The Midpoint Formula

The midpoint of the segment that joins points (x1,y1) and (x2,y2) is the point

2yy

,2

xx 2121

•

•

(-4,2)

(6,8)

282

,2

64- •(1,5)

Exercises

3. M (3,5) A (0,1) B (x,y)

(6,9)

This is the midpoint

To find the coordinates of B:

x-coordinate:

3 = 0 + x2

6 = 0 + xx = 6

y-coordinate:

5 = 1 + y2

10 = 1 + yy = 9

II. Standard Form: (Ax + By = C)II. Standard Form: (Ax + By = C). Getting x and y intercepts: (x, 0) . Getting x and y intercepts: (x, 0) and (0, y)and (0, y)

1) 2x + 3y = 61) 2x + 3y = 6

20

3 0

Try the cover up method!!!

.(0, 2)

.(3, 0)

yx

14

2y x

II. Slope-Intercept Form (y = mx + b): m = slope; b = y-intercept

y = 2

.(0, 4). ..

..

yorizontal

Why?

Thus y=2!!

.(-1, 2) .(6, 2).(-6, 2)

III. Finding Slope-Intercept Form: (y = mx + b)

3x – 4y = 10

m = _____ b = _____

-3x -3x

-4y = -3x + 10-4 -4 -4

y = 3/4x – 5/2

3/4 -5/2

IV. Systems of Equations: Two lines in a coordinate plane can do two things: (1) intersect (perpendicular or not) (2) not intersect

(parallel)

Systems Algebraic Graph

By Substitution2x + y = 8y = 2x

Isolate a variable first.This is already done.Then substitute.

( )( ) 2x + (2x) = 8

4x = 8x = 2

Substitute 2 back in for x in the easier equation!!

y = 2x

y = 2(2)

y = 4

The solution to the system is (2, 4)

Graph 2x + y = 8 -2x -2x

y = -2x + 8

y = -2x + 8

Graph y = 2x

y =

2x

.(2,4)

IV. Systems of Equations: Two lines in a coordinate plane can do two things: (1) intersect (perpendicular or not) (2) not intersect

(parallel)

Systems Algebraic GraphBy Addition w/Multiplication

2x + y = 63x – 2y = 2

Graph 2x + y = 6 -2x -2x

y = -2x + 6

y =

3/2x

– 1

Graph 3x – 2y = 2

y = -2x + 6

.(2,2)

7x = 14

x = 2Substitute 2 back in for x in the easier equation!!

4(2) + 2y = 12

8 + 2y = 12

2y = 4

y = 2

The solution to the system is (2, 2)

-8 -8

-3x -3x -2y = -3x + 2 -2 -2 -2

y = 3/2x – 1

( )24x + 2y = 12

Given x and y intercepts:

1. x-int: 2 y-int: -3

(2,0) (0,-3)

●

●

Notice that the slope is

rise 3

run 2or

(2,0)

(0,-3)

(-3)

2

or y-int

x-int.

The y intercept (b) of -3 is given

The equation in slope intercept form isy = 3

2x - 3

-

opposite

Given InterceptsTo write the equation in slope-intercept form use the pattern :

y = y-intercept

x-interceptx + y-intercept

slope m b

7 2 5

4 1 3m

52 ( 1)

3y x

Step 1: Compute slope

Step 2: Use PS Form

Step 3: Simplify to SI Form

+2y = 5/3x + 1/3

Using (1, 2)

Part IV #1: Given 2 points. (1,2) and (4,7)

5 52

3 3y x

6

3

You can check with other point:

7 = 5/3(4) + 1/3

7 = 20/3 + 1/3

7 = 21/3

7 = 7 check!

x = 8

Part VI #5:

(8,7) and parallel to x = -2

x = 2

Part VI #6:

(2,2) and perpendicular to y = 3

All vertical lines are parallel

A vertical line is perpendicular to a horizontal line

• Chapter 13 WS• How can you get 100% on your final?

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