Midpoints! Midpoint on a Number Line Midpoint in the Coordinate Plane Derivation of the Midpoint Formula Fractional Distance!

Midpoints! Midpoint on a Number Line | Midpoint in the Coordinate Plane | Derivation of the Midpoint Formula | Fractional Distance! Midpoints! Learn...
Author: Elizabeth Snow
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Midpoints! Midpoint on a Number Line | Midpoint in the Coordinate Plane | Derivation of the Midpoint Formula | Fractional Distance!

Midpoints! Learning Objectives! •  Determine the coordinates of a midpoint from the two endpoints! •  Adapt the midpoint formula for fractions of the distance other than one-half!

Midpoint on a Number Line! !

•  Midpoint – the point in the in the middle of a line segment! •  Bisector – a point, line, or line segment that creates a division of a geometric object into two congruent parts! •  A line has no endpoints and no midpoint! •  Line segments have midpoints! –  Ex) The line segment from 3 to 7 on a number line has a midpoint at 5! –  Midpoint coordinate – the average of the endpoints coordinates!

Midpoint on a Number Line! Example! !

Ex) The midpoint of line segment AB is the point M, where AM = 2x + 1 and MB = 4x – 7. Find the length of line segment AB . !

Analyze! Formulate ! !

!

Justify! ! ! !AB = 2(AM)!

!! AM ≅ MB

Determine! ! ! !2x + 1 = 4x – 7! ! ! !

!

! ! !!

!4 = x! !AM = 2x + 1 = 2(4) + 1 = 9!

( )

AB = 2 AM = 2 ( 9 ) = 18

!

!

!AM = 9!

Evaluate!

Midpoint in the Coordinate Plane! ! •  Endpoints have both an x- and y-value in a twodimensional coordinate plane!

•  Midpoint formula – the formula connecting the coordinates of the midpoint with the coordinates of the endpoints !

The midpoint formula!

Given points A(x1,y1) and B(x2,y2), ! ⎛ x1 + x!2 y1 + y2 ⎞ ⎜⎝ 2 ! , 2 ⎟⎠ ! is the midpoint of the line segment AB . !

Midpoint in the Coordinate Plane! ! •  Ex) Find the midpoint of line segment RS with endpoints R(2,1) and S(9,6)! ⎛ x1 + x2 y1 + y2 ⎞ ⎛ 1 + 9 2 + 6 ⎞ ⎜⎝ 2 , 2 ⎟⎠ = ⎜⎝ 2 , 2 ⎟⎠

⎛ 10 8 ⎞ =⎜ , ⎟ ⎝ 2 2⎠ = ( 5, 4 )

Midpoint in the Coordinate Plane! Example! !

Ex) The midpoint of line segment AB is the point M(4,7). The coordinates of A are (–  1,1). Find the coordinates of B.! ! Analyze! ! !One endpoint and midpoint are !

! provided!

Formulate ! ! ! ⎛ x1 !+ x2!! y1 + y2 ⎞ ⎜⎝ 2 , 2 ⎟⎠ = ( 4,7 ) ! Determine!

−1 + x2 =4 2

1 + y2 =7 2

−1 + x2 = 8

1 + y2 = 14

x2 = 9

y2 = 13

Justify! !! !Coordinates of B are (9,13)! Evaluate! ! !Substituting values into ! ! ! !

!

!midpoint formula provided an !analytical way to find the ! ! !coordinates of B!

Derivation of the Midpoint Formula! ! •  Midpoint – the average of the coordinates of the endpoints! –  The x-coordinate – the ! average of the horizontal ! components! –  The y-coordinate – the ! average of the vertical ! components!

Derivation of the Midpoint Formula! ! •  Midpoint formula ! ! !

⎛ x1 + x2 y1 + y2 ⎞ ⎜⎝ 2 , 2 ⎟⎠ –  Distance an object moves from one point to another! –  Object moves half the horizontal distance and half the vertical distance between A and B!

Derivation of the Midpoint Formula! ! •  Horizontal displacement (∆ x) = x2 – x1! •  Vertical displacement ( ∆ y) = y2 – y1! 1 xm = x1 + ( ∆ x ) 2 1 xm  = x2 − x1 )     x1 + ( 2       1 1 xm = x1 + x2 − x1 2 2            

1 1 xm = x1 − x1 + x2 2 2      

1 1 x1 + x2 2 2       1 xm = ( x1 + x2 ) 2

xm =

Add one-half of the horizontal displacement to the x-coordinate of A.! Replace ∆ x with x2 – x1! ! Distribute the 1/2! ! Commute the second and third terms! ! One whole x minus one-half x is one-half x! Factor the 1/2!

Derivation of the Midpoint Formula! ! •  Result of moving half the vertical distance !

1 1 ym = y1 + ( ∆ y ) = ( y1 + y2 ) ! 2 2 •  The midpoint is the location halfway from A to B along AB !

Fractional Distance! •  Distance from points on line segment AB ! –  Add the fraction of the displacement from A to B to the coordinate(s) of A !

•  Fractional distance – the length of a designated portion of a line segment! –  Ex) Displacement from point P to point Q on a number line is 13 – 1 = 12 ! •  If an object is moved one-third of that displacement, then it moves four units!

Fractional Distance! !

•  Ex) Displacement from point Q to point P on a number line is 1 – 13 = –12 ! –  One-third of the displacement is –4! –  New location of the object is at 13 – 4 = 9! –  Fractional displacement is –4!

Fractional Distance! ! •  Fractional distance in the coordinate plane requires computing x- and y-coordinates separately !

•  Ex) Consider points A(x1,y1) and B(x2,y2) where an object moves from B to A! –  Horizontal displacement ! !x1 – x2 ! –  Vertical displacement! !y1 – y2

Fractional Distance Example! ! Ex) Given points D(–2,10) and E(10,1), determine the coordinates of a point 10 units from E on the line segment DE. ! ! Analyze! ! Formulate ! a x2 − x1 ) ( b ! a yr = y1 + ( y2 − y1 ) b ! Determine! ! !

!!

xr = x1 +

DE =

( −2 − 10) + (10 − 1)

DE =

( −12)

2

2

+ 92

2

DE = 144 + 81 DE = 225

DE = 15

10 2 = 15 3

Fractional Distance Example! ! Ex) Given points D(–2,10) and E(10,1), determine the coordinates of a point 10 units from E on the line segment DE. ! Justify! ! Determine! ! !Coordinates of a point 10 ! ! !! a xr = x1 + ( x2 − x1 ) b 2 xr = 10 + ( −2 − 10 ) 3 xr = 10 +

2 −12 ) ( 3

a yr = y1 + ( y2 − y1 ) ! ! b

2 yr = 1 + (10 − 1) 3 2 yr = 1 + ( 9 ) 3

xr = 10 − 8

yr = 1 + 6

xr = 2

yr = 7

! ! !

! !units from E required ! !finding the length of the ! line segment (15) and what !fraction of the line segment !is 10 units long (2/3)!

Evaluate !! ! !Result can be checked !

!using the distance formula !

Fractional Distance Example! ! Ex) The point R(–1,0) is one-fifth of the way from point P(– 3, –2) on the line segment PQ. What are the coordinates of point Q?! Analyze! a Formulate x! r = x1 + ( x2 − x1 )

b a ! yr = y1 + ( y2 − y1 ) b Determine ! !

(

1 x − ( −3 ) 5 2 1 −1 = −3 + ( x2 + 3 ) 5

−1 = −3 +

1 x2 + 3 ) ( 5 10 = x2 + 3 7 = x2 2=

)

Justify!

Evaluate !! ! ! !If PR = 1/5PQ, then ! ! !5(PR) = PQ and ! 1 0 = −2 + ( y2 − ! ( −2!)) !5(y2 – y1 ) = y2 – y1 ! 5 1 0 = −2 + ( y2 + 2 ) 5 1 2 = ( y2 + 2 ) 5 10 = y2 + 2 8 = y2

!

Construction - Segment Bisector ! !

Midpoints! Learning Objectives! •  Determine the coordinates of a midpoint from the two endpoints! •  Adapt the midpoint formula for fractions of the distance other than one-half!

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