1 3 distance and midpoint
Transcript of 1 3 distance and midpoint
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Section 1-3Section 1-3Jim Smith JCHSJim Smith JCHS
Spi.2.1.ESpi.2.1.E
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The distance between 2 points isThe distance between 2 points isthe absolute value of the difference the absolute value of the difference
of the coordinatesof the coordinates..
The distance between exit 417 and 407 is
| 417 – 407 | = 10 or
| 407 – 417 | = | -10 | = 10
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The distance between A and B isThe distance between A and B is
| | | | | | | | | | | | | | | | | | | | | | | | | | | |
-5 4-5 4
A BA B
| -5 – 4 | = | -9 | = 9
Our distances should alwaysOur distances should always be positivebe positive
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| | | | | | | | | | | | | | | | | | | | | | | | | | | | -6 12-6 12
A BA B
The midpoint of a segmentThe midpoint of a segmentis the average of the is the average of the
coordinatescoordinates
-6 + 12 2
= 62
= 3
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Review GraphingReview Graphingyy
xx
( 0,0 )( 0,0 )OriginOrigin
PositivePositive
NegativeNegative
Order ( X,Y )Order ( X,Y )
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AA
BB
The Distance Formula Is Derived The Distance Formula Is Derived From The Pythagorean FormulaFrom The Pythagorean Formula
66
1515
66² + 15² = AB²² + 15² = AB²
√√261261
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Distance FormulaDistance Formula
Dist = ( x - x )Dist = ( x - x )² + ( y - y )²² + ( y - y )²
Remember the order ( x , y )Remember the order ( x , y )
Check yourself …Check yourself … our answers should be positiveour answers should be positive
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Find the distance between:Find the distance between:
( 3 – 8 )( 3 – 8 )² + ( 6 - 10 )²² + ( 6 - 10 )²
( -5 )² + ( -4 )²( -5 )² + ( -4 )²
25 + 1625 + 16
41 = 6.4041 = 6.40
( 8 – 3 )( 8 – 3 )² + ( 10 – 6 )²² + ( 10 – 6 )²
( 5 )² + ( 4 )²( 5 )² + ( 4 )²
25 + 1625 + 16
41 =6.4041 =6.40
( 3 , 6 ) and ( 8 , 10 )( 3 , 6 ) and ( 8 , 10 )
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MIDPOINTMIDPOINTThe midpoint of a segment is half way The midpoint of a segment is half way
between the x’s and half way between the y’sbetween the x’s and half way between the y’sYou can call it the average You can call it the average
66
1010
MidpointMidpoint
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Midpoint FormulaMidpoint FormulaX + X , Y + YX + X , Y + Y 2 22 2Find the midpoint ofFind the midpoint of
( 2,8 ) and ( 6,4 )( 2,8 ) and ( 6,4 )
2 + 6 , 8 + 4 = 8 ,12 = ( 4 , 6 )2 + 6 , 8 + 4 = 8 ,12 = ( 4 , 6 ) 2 2 2 2 2 2 2 2
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XX11 + X + X22 2 2
== XXMIDMIDYY11 + Y + Y22
22= = YYMIDMID
What If We Knew The Midpoint Of A SegmentWhat If We Knew The Midpoint Of A SegmentAnd One Endpoint? How Would We Find TheAnd One Endpoint? How Would We Find TheOther Endpoint?Other Endpoint?
Think Of The Formula As:Think Of The Formula As:
Endpoints MidpointEndpoints Midpoint
(( XX11 , , YY11 )) (( XX22 , , YY22 )) ( ( XXmidmid , , YYmidmid ))
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Endpoint ( 3 , 5 ) Midpoint ( 6 , -2 ) Endpoint ( 3 , 5 ) Midpoint ( 6 , -2 ) Find The Other Endpoint.Find The Other Endpoint.XX11 + + XX22 22
== XXMIDMID YY11 + Y + Y22
22= = YYMIDMID
Find ( Find ( XX2 2 ,,YY22 ) )
3 + X3 + X22
223 + X3 + X22
XX22 = 9 = 9
= 6= 6
= 12= 12
5 + Y5 + Y22
225 + Y5 + Y22
YY22 = -9 = -9
= -2 = -2
= -4= -4
( 9 , -9 )( 9 , -9 )