Edited Per4 Analytic Geometry

Analytical Geometry By: Kim Caligagan , Franca Petrelli, and Alan Merluza

table of contents Distance Formula Midpoint Formula Slope Formula Graphing Lines -table of values - x and y intercept -slope y intercept Equation of a line

DISTANCE FORMULA (X 1 ,Y 1 ) (X 2 ,Y 2 ) d=√ (y2-y1)2 + (x2-x1)2

Distance Formula

d=√ (y 2 -y 1 ) 2 + (x 2 -x 1 ) 2

For example, consider the two points A (2,5) and B (5,1),

so: x1 = 2, y1 = 5, x2 = 5, and y2 = 1.

Substituting into the distance formula we have:

d= √(1-5) 2 +(5-2) 2

d= √(4) 2 +(3) 2

d =√16+9

d =√25

d =5

ALWAYS REMEMBER THE FORMULA OF DISTANCE THEN SUBSTITUTE THE GIVEN POINTS!!!

Midpoint Midpoint MIDPOINT FORMULA: X 2 X + 1 ---------------- 2 , Y Y 2 1 + ---------------- 2

EXAMPLE: FIND THE MIDPOINT BETWEEN (7,3) AND (5,2) USING THE MIDPOINT FORMULA SUBSTITUTE THE GIVEN POINTS IF X1=7,Y1=3 AND X2=5,Y2=2 X 2 + X 1 ------------ 2 , Y + Y 1 ------------ 2 5+7 ---------- 2 = 12 ---------- 2 = 6 (X) 2+1 ---------- 2 = 2 ---------- 3 (Y)

LETS TRY ANOTHER QUESTIONS:

THE LINE SEGMENT FROM B (1,-3)TO C(H,K)HAS ITS MIDPOINT AT (4,5)FIND THE H AND K .

SO YOU NOTICED THAT THE MIDMOINT IS GIVEN ..DONT WORRY JUST FOLLOW THE FORMULA OF MIDPOINT;

FIRST,IDENTIFY THE NUMBER AS SHOWN BELOW

B(X 1 =1 and Y 1 = -3) MP(X M =4 and Y M =5) C(X 2 = H and Y 2 = K )

SO THE MIDPOINT OF THE PREVIOUS SLIDE IS (6,3/2) MP = X1+X2 2 USED THE INFORMATION ABOVE THEN SUBSTITUTE IT TO THE GIVEN FORMULA 4= 1+H 2

STEP 2 : IN ORDER TO ELIMINATE THE 2 JUST SIMPLY MULTIPLY IT ON THE BOTH SIDES

8=1+H STEP 3 : Transpose 1 to the left side

8-1=H STEP 4: YOU SHOULD ALWAYS CHANGE THE SIGN EVERYTIME YOUR TRANPOSING A CERTAIN NUMBER.

4= 2 ( 1+ H 2 ) 2 MP = 2 Y1 +Y2 5= -3+K 2 2 ( 5= -3+K 2 ) 2 10= -3+K 10+3=K 13=K 7=H by kimberly

slope formula Y2-Y1 X2-X1

Here is an example of how to do slope formula:

-The slope m passes through the points of (2, 4) and (3, 5)

-Label the points: x1 y1 x2 y2

( 2 , 4 ) (3 , 5)

Using slope formula: y2 – y1 5-4 1

x2 – x1 3-2 1

That means that the slope m is 1

SLOPE FORMULA #2

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Graphing lines Table of Values X and Y intercept Slope Y intercept

The Table of values system shows what you could get for Y, if X is a certain number. For example:

Y = X – 9  Lets say we

wanted X to be

-1.

Y= X – 9

Y= (-1) – 9

Y=-10

* You can put any number for X.

Table of Values X Y X -1 -10 0 -9 1 -8

There are different equations you can use to receive your answer:

Ax+By+C=0  Standard form.

Y=mx+b  Slope Y-intercept form.

Y-y1=m(x-x1) Point-slope form

Ax+By=C

The X-intercept happens when the y-intercept equals to zero. For example:

3x+8y+24=0

3x+8(0)+24=0  *The zero takes the place of the Y or X

3x+24=0

3 3

X= 8

The Y-intercept happens when the X- intercept equals to zero. For example:

5x+2y+18=0

5(0)+2y+18=0

2y + 18=0

2 2

Y=9

X and Y intercept

For Slope Y-intercept we use the equation: Y=mx+b

If your equation is Y=1/3x+4

Then you will put this on a graph by labeling it as 1/3 would be your rise over run . Your Y (4) will be put over one. So, it will become:

M=1 Y=4 Now, you are going to graph

3 1 these points.

Slope Y-intercept slope y-intercept

Graphing your points First, you are going to rise 4,now that you have that point from there you are going to rise one and run three. This is you rising 4. This is you rising 1. And this is you running 3.