1.3 Distance and Midpoints
Objective and Standard
• Review distance and midpoint formulas,
introduce use of compass to copy and bisect line
segments..
• Check.3.4 Apply the midpoint and distance formulas to points and
segments to find midpoints, distances, and missing information in
two and three dimensions.
•
Check.4.3 Solve problems involving betweeness of points and
distance between points (including segment addition).
Review
• How long is a line?
– infinity
• Line segment can be measured because
– it has two endpoints.
• .
Congruence
• Two segments having the same measure
are congruent.
A
4 in
• Indicated by red slash
Z
4 in
U
B
Constructions – Equal Lines
• What is the precision of the lines you drew?
• Use Compass to create a new segment MN
that is congruent to TU.
– Draw a line on your paper. Draw point M.
– Place compass point a T, adjust the setting so the
pencil is at point U.
– Place point at M, draw and arch that intersects the
line and label N
• Using compass to create a line segment XZ
= XY + YZ
Constructions - Midpoints
• Calculate the midpoint of XZ
• Use compass to calculate
midpoint of XZ, label as M.
– Place the compass at X
– Adjust settings so the width is greater
that ½ XZ.
– Draw arcs above and below XZ
– Using same compass settings, place
compass point at Z and repeat.
– Find intersections of the arcs
– Using straight edge connect the two
points
– Label intersection of point as M
• Measure segment XM compare
against your calculation.
X
M
Z
Distance
• Calculate Distance on number line
– Absolute value of difference between two
points.
15
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
Distance Coordinate Grid
Find the distance between
R(5,1) and S(-3,-3)
Distance Formula
R
S
R(x1, y1) x1 =5, y1 =1
S(x2, y2) x2 = -3, y2 = -3
Distance Coordinate Grid
Find the distance
between E(-4,1) and
F(3, -1 )
E
F
Distance Formula
Midpoint Exercise
Y
Z
B
X
C
A
• What are the coordinates of
point C?
• (2. 5)
• What are the lengths of AC and
CB?
• 3 units each
• What are the coordinates of Z?
• (-1, 5)
• What are the lengths of XZ and
ZY?
• Between 3 and 4 units each
What sort of rule did you write for XYZ?
How would you write a rule for ABC?
Midpoint of Line Segment
Number Line
Coordinate Plane
Endpoints at A and B
Endpoints at (x1, y1) and (x2, y2)
Midpoint = (A + B)
2
Midpoint = (x1+x2) , (y1+y2)
2
2
Midpoint = (A + B)
2
Calculate Midpoint
-12
J
16
M
K
• Find the Midpoint of JK
-12 + 16
2
4
2
=2
• Find the Length of JK
|-12 + 16|
4
Midpoint = (x1+x2) , (y1+y2)
2
2
Calculate Midpoint
R
S
Find the midpoint between
R(5,1) and S(-3,-3)
Midpoint = (5+(-3)) , (1+(-3))
2
2
2,
-2
2
2
(1, -1)
Calculate Endpoint
Find the coordinates of X if Y(-2, 2) is the midpoint of
XZ and Z has the coordinates of (2, 8)
Let Z = (x2, y2) in the formula
Y(-2,2) =
(x1+2) , (y1+8)
2
2
-2 = (x1+2)
2
-4 = x1+2
-6 = x1
Solve each problem separately
2=
(y1+8)
2
4 = y1+8
-4 = y1
Calculate Measurements
11 + 2x
4x - 5
A
C
B
Find BC if B is the midpoint of AC
AB = BC
4x – 5 = 11 + 2x
2x = 16
x=8
Assignment – Block Geometry
• On a separate sheet of paper – turn in before
you leave
• Create a line segment AB, of unknown length
– Create line EF, of unknown length
– Use compass to find midpoint of line EF and label it M
– Create third line equivalent to sum of line AB and EM.
– Page 31, 12 - 56 every 4th