The Distance and Midpoint Formulas

By: L. Keali’i Alicea

Geometry Review!

• What is the difference between the What is the difference between the symbols AB and AB?symbols AB and AB?

Segment ABSegment AB

The The lengthlength of of Segment ABSegment AB

The Distance Formula

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

212

212 )()( yyxxd

Find the distance between the two points.

• (-2,5) and (3,-1)(-2,5) and (3,-1)• Let (xLet (x11,y,y11) = (-2,5) and (x) = (-2,5) and (x22,y,y22) = (3,-1)) = (3,-1)

22 )51())2(3( d

3625d

81.761d

Classify the Triangle using the Classify the Triangle using the distance formula (as scalene, distance formula (as scalene,

isosceles or equilateral)isosceles or equilateral)

29)61()46( 22 AB

29)13()61( 22 BC

23)63()41( 22 AC

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

C: (1.00, 3.00)

B: (6.00, 1.00)

A: (4.00, 6.00)

C

B

A

The Midpoint Formula

• The midpoint between the two The midpoint between the two points (xpoints (x11,y,y11) and (x) and (x22,y,y22) is:) is:

)2

,2

( 1212 yyxxm

Find the midpoint of the segment Find the midpoint of the segment whose endpoints are (6,-2) & (2,-9)whose endpoints are (6,-2) & (2,-9)

2

92,

2

26

2

11,4

Write an equation in slope-intercept Write an equation in slope-intercept form for the perpendicular bisector form for the perpendicular bisector

of the segment whose endpoints are of the segment whose endpoints are C(-2,1) and D(1,4).C(-2,1) and D(1,4).

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

• Now, find the slope of CD.Now, find the slope of CD. m=1m=1

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

Assignment10.1 A (all)10.1 A (all)

10.1 B (2-14 even, 15-18)10.1 B (2-14 even, 15-18)