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!