8.1 The Distance and Midpoint 8.1 The Distance and Midpoint FormulasFormulas

p. 490p. 490

What is the distance formula?What is the distance formula?

How do you use the distance formula to How do you use the distance formula to classify a triangle?classify a triangle?

What is the midpoint formula?What is the midpoint formula?

How do you write the equation for a How do you write the equation for a perpendicular bisector given two points?perpendicular bisector given two points?

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

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

81.761 d

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

C: (1.00, 3.00)

B: (6.00, 1.00)

A: (4.00, 6.00)

C

B

A

2. The vertices of a triangle 2. The vertices of a triangle are are RR(– 1, 3)(– 1, 3), , SS(5, 2)(5, 2), and , and TT(3, 6)(3, 6). Classify . Classify ∆∆RSTRST as as scalenescalene, , isoscelesisosceles, or , or equilateralequilateral..

SOLUTIONSOLUTION

STST== (3 – 5)(3 – 5)22 + (6 – 2) + (6 – 2)2 2 = 20= 20

TRTR == (–1 –(–3)(–1 –(–3)22 + (3 – 6) + (3 – 6)2 2 = 25 = 25

= 5= 5

RSRS== (5 – (–1)(5 – (–1)22 + (2 – 3) + (2 – 3)2 2 = 36 = 6= 36 = 6

ANSWERANSWER Because Because RS ≠ ST ≠ TR,RS ≠ ST ≠ TR, so so RSTRSTis an scalene triangle.is an scalene triangle.

R –1,3R –1,3

S S 5,25,2

TT 3,6 3,6

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

LetLet ( ( xx11, , yy1 1 ) = (–5, 1)) = (–5, 1) andand ( ( xx22, , yy2 2 ) = (– 1, 6 ).) = (– 1, 6 ).

Find the midpoint of the line Find the midpoint of the line segment joining segment joining (–5, 1)(–5, 1) and and (–1, 6)(–1, 6)..

SOLUTIONSOLUTION

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

SOLUTIONSOLUTION

STEPSTEP 11 Find the midpoint of the Find the midpoint of the line segment.line segment.

Write an equation for the Write an equation for the perpendicular bisector of the line perpendicular bisector of the line segment joining segment joining AA(– 3, 4)(– 3, 4) and and BB(5, 6).(5, 6).

==( )( )– – 3 + 5 4 + 6 3 + 5 4 + 6

22 22 ,,( )( )xx11 + + xx22 yy11 + + yy22

22 22,, = (1, 5)= (1, 5)

STEPSTEP 22

mm = =yy22 – – yy11

xx22 – – xx11==

6 – 46 – 45 – (– 3)5 – (– 3)

== 2288

== 1144

STEPSTEP 33 Find the slope of the perpendicular bisector:Find the slope of the perpendicular bisector:

–– 11mm

––== 1 1 1/41/4 = – 4 = – 4

Calculate the slope ofCalculate the slope of ABAB

ANSWERANSWER An equation for the An equation for the perpendicular bisector ofperpendicular bisector of ABAB isis y y = – 4= – 4x x + 9.+ 9.

STEPSTEP 44

Use point-slope formUse point-slope form::

y y – 5 = – 4(– 5 = – 4(x x – 1), – 1), y y = – 4= – 4x x + 9.+ 9.oror

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

Asteroid CraterAsteroid Crater

Many scientists believe that an asteroid slammed into Many scientists believe that an asteroid slammed into Earth about Earth about 6565 million years ago on what is now million years ago on what is now Mexico’s Yucatan peninsula, creating an enormous Mexico’s Yucatan peninsula, creating an enormous crater that is now deeply buried by sediment. Use the crater that is now deeply buried by sediment. Use the labeled points on the outline of the circular crater to labeled points on the outline of the circular crater to estimate its diameter. (Each unit in the coordinate estimate its diameter. (Each unit in the coordinate plane represents plane represents 11 mile.) mile.)

See page 492

STEP 1STEP 1

Write equations for the perpendicular bisectors of Write equations for the perpendicular bisectors of AOAO and and OBOB using the method of Example using the method of Example 44..

y y = – = – x x + 34+ 34 Perpendicular bisector of Perpendicular bisector of AOAO

y y = 3= 3x x + 110+ 110 Perpendicular bisector of Perpendicular bisector of OBOBSTEP 2STEP 2Find the coordinates of the center of the circle, where Find the coordinates of the center of the circle, where AOAO and and OBOB intersect, by solving the system formed by the two intersect, by solving the system formed by the two equations in Step equations in Step 11..

y y = – = – x x + 34+ 34 Write first equation.Write first equation.

33x x + 110 = – + 110 = – x x + 34 + 34 Substitute for Substitute for yy..

44x x = – 76= – 76 Simplify.Simplify.

x x = – 19= – 19 Solve for Solve for xx..

y y = – (– 19) + 34= – (– 19) + 34Substitute the Substitute the xx-value into the -value into the first equation.first equation.

y y = 53= 53 Solve for Solve for yy..

The center of the circle isThe center of the circle is C C (– 19, 53).(– 19, 53).

STEP 3STEP 3

Calculate the radius of the circle using the distance Calculate the radius of the circle using the distance formula. The radius is the distance between formula. The radius is the distance between CC and any and any of the three given points.of the three given points.

OC OC = (–19 – 0)= (–19 – 0)22 + (53 – 0) + (53 – 0)22 = 3170 56.3 = 3170 56.3

UseUse ((xx11, , yy11) = (0, 0)) = (0, 0) andand ((xx22, , yy22) = (–19, 53).) = (–19, 53).

ANSWERANSWER

The crater has a diameter of aboutThe crater has a diameter of about 2(56.3) = 112.62(56.3) = 112.6 miles.miles.

For the line segment joining the two given points, (a) find the For the line segment joining the two given points, (a) find the midpoint and (b) write an equation for the perpendicular midpoint and (b) write an equation for the perpendicular bisector.bisector. 5. 5. (3, 8), (–5, –10)(3, 8), (–5, –10)

SOLUTIONSOLUTION

3 + (– 5) 8 + (–10) 3 + (– 5) 8 + (–10) ( )( )== 22 22

,,( )( )xx11 + + xx22 yy11 + + yy22

22 22

,,

LetLet ((xx11, , yy1 1 ) = (3, 8)) = (3, 8) andand ( ( xx22, , yy2 2 ) = (– 5, –10).) = (– 5, –10).

midpoint ismidpoint is (–1 , –1) (–1 , –1)

STEPSTEP 22 Calculate the slopeCalculate the slope

mm = =yy22 – – yy11

xx22 – – xx11==

––10 – 810 – 8

– –5 – 35 – 3==

––1818– – 88

==9944

STEPSTEP 33 Find the slope of the perpendicular bisector:Find the slope of the perpendicular bisector:

––11mm ==

4499

– – 11== 99

44STEPSTEP 44 Use point-slope formUse point-slope form::

y y 1 = 1 = ( (x x 1), 1), 4499

y y = = x x oror 4499

1313 99

• What is the distance formula equation?What is the distance formula equation?

• How do you use the distance formula to classify a How do you use the distance formula to classify a triangle?triangle?

The distance formula will tell you the length of the sides of The distance formula will tell you the length of the sides of the triangle. (2= isosceles, 3=equilateral)the triangle. (2= isosceles, 3=equilateral)

• What is the midpoint formula?What is the midpoint formula?

• How do you write the equation for a perpendicular bisector How do you write the equation for a perpendicular bisector given two points?given two points?

Use the points to find the slope, use the negative reciprocal, Use the points to find the slope, use the negative reciprocal, use the midpoint formula to find a the middle point and use use the midpoint formula to find a the middle point and use y = mx+b to write your equationy = mx+b to write your equation

212

212 )()( yyxxd

)2

,2

( 1212 yyxxm

8.1 Assignment8.1 Assignment

p.p. 493493

3-36 every 33-36 every 3rdrd problem problem