10.1 The Distance and Midpoint Formulas 10.1 The Distance and Midpoint Formulas

What is the difference between the symbols AB and AB?

Segment ABSegment AB

The The lengthlength of of Segment ABSegment AB

The Distance d between the points (x1,y1) and (x2,y2) is :

212

212 )()( yyxxd

A

B

C

How long is a? 4

How long is b? 4

Pythagorean Formula:

A2 + b2 = c2 or c = √(a2 +b2)

How long is c? 5.7

A = x2 – x1 and B = y2 –y1

(-2,5) and (3,-1) Let (x1,y1) = (-2,5) and (x2,y2) = (3,-

1)22 )51())2(3( d

3625d

81.761d

29)61()46( 22 AB

29)13()61( 22 BC

23)63()41( 22 AC

C: (1.00, 3.00)

B: (6.00, 1.00)

A: (4.00, 6.00)

C

B

A

Because AB=BC the triangle is Because AB=BC the triangle is ISOSCELESISOSCELES

The midpoint between the two points (x1,y1) and (x2,y2) is:

)2

,2

( 1212 yyxxm

2

92,

2

26

2

11,4

First, find the midpoint of CD. (-1/2, 5/2)

Now, find the slope of CD. m=1

* Since the line we want is perpendicular to the given segment, we will use the opposite reciprocal slope for our equation.

(y-y(y-y11)=m(x-x)=m(x-x11) or y=mx+b) or y=mx+b

Use (xUse (x11 ,y ,y11)=(-1/2,5/2) and m=-1)=(-1/2,5/2) and m=-1

(y-5/2)=-1(x+1/2) or 5/2=-1(-1/2)+b(y-5/2)=-1(x+1/2) or 5/2=-1(-1/2)+b

y-5/2=-x-1/2 or 5/2=1/2+by-5/2=-x-1/2 or 5/2=1/2+b

y=-x-1/2+5/2 or 5/2-1/2=by=-x-1/2+5/2 or 5/2-1/2=b

y=-x+2 or 2=by=-x+2 or 2=b

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

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