Download Discovering the Midpoint Formula Spring 2013

Discovering the Midpoint Formula
The midpoint formula is used in Geometry to determine the coordinates of a point at the exact middle point of a line segment. If
given the two endpoints of a line segment, you should be able to determine the coordinates of the midpoint.
Definition: Every midpoint of a line segment is a point that bisects a line segment into two congruent parts.
1. Reverse this definition (paying attention to the number of underlines in each part of the definition):
Every
Midpoint of a line
segment:
that
is a
.
2. Consider the graph below, which shows two points, B and E, on a coordinate plane. Line segment
midpoint
BE has its endpoints at point B (4, 3) and E (8,11) . It may help to think of point B as the
"Beginning," and point E as the "End." We are interested in finding the coordinates of the midpoint,
point M.
E
a) Using a straightedge,
draw line segment BE .
10
c) Explain what strategy you
used to find the midpoint:
b) Using a ruler, folding, or
some other strategy, locate the
midpoint. Label it point M.
d) Our goal is to find an
algebraic rule to find the
coordinates of the midpoint.
To start, write down the
x-coordinates of the endpoints
of the segment:
Point B x-coordinate:
5
Point E x-coordinate:
e) Now, write down the
x-coordinate of the midpoint:
B
Point M x-coordinate:
f) Using addition, subtraction,
and/or division by 2, find a
simple way to combine the
numbers from part (d) to get
the answer from part (e):
5
g) Let's write a formula for
the x-coordinate of the
midpoint:
Write an equation that begins
"6 = …." where the "…"
portion incorporates your idea
from part (f):
h) Let's rename point B as
point 1, and point E as point 2.
To rename point B as point 1,
we write this:
B(4, 3)  ( x1, y1 )
To rename point E as point 2,
write it like this: (fill in the
blanks):
E (8,11)  (
k) Write down the
y-coordinate of the midpoint:
Point M y-coordinate:
,
i) Let's write the formula for
the x-coordinate of the
midpoint. You will combine
the ideas of parts (f, g, and h)
to write the formula:
Starting with your equation in
(g), replace the 6 with xm ,
x1 , and
replace the 8 with x2 :
j) Now, let's turn our attention
to the y-coordinates: Write
down the y-coordinates of the
endpoints of the segment:
Point B y-coordinate:
Point E y-coordinate:
replace the 4 with
)
l) Using addition, subtraction,
and/or division by 2, find a
simple way to combine the
numbers from part (j) to get
the answer from part (k):
m) Let's write the formula
for the y-coordinate of the
midpoint:
Write an equation that begins
"7 = …." where the "…"
portion incorporates your idea
from part (l):
n) Let's write the formula for
the y-coordinate of the
midpoint: Starting with your
formula in part (m), replace
the 7 with ym , replace the 3
y1 , and replace the 11
with y 2 :
with
x1  x2
y  y2 
, ym  1
 . Let's use it on
2
2 
the following problem: "Find the midpoint of the segment with its endpoints at A(5,  7) and B (9, 23) ."


3. The midpoint formula, as you'll find it in textbooks, is typically written as  xm 
a) Let's label point A as point 1, like this,
with little labels above each number:
x1
A (
5
y1
,
7
b) Do the same thing for point B, but
label it as point 2, not point 1: (fill in the
boxes)
 
)
B
d) Substitute the known values
x1 , x2 , y1 , and y2 into the formula:
4. What is the midpoint of
(
9
c) Copy down the midpoint formula:
 
,
23
)
e) Simplify:
f) So, what is the midpoint of AB with
endpoints A(5,  7) and B (9, 23) ?
Answer as an ordered pair:
CD with endpoints C (2,18) and D(5,  22) ? Express your answer as an ordered pair.
5. As is always the case in math, you can go forwards, and you can go backwards…. What if we know one endpoint of segment
AB is point B (4,  10) , and the midpoint is at M (5, 3) : How do we figure out the coordinates of the other endpoint?
a) Let's label point B as point 1, like this,
with little labels above each number:
x1
B
(
4
y1
,
b) Do the same thing for the
midpoint M: (fill in the boxes with
xm and ym :
 
 10 )
M
(
5
c) Copy down the midpoint formula:
 
,
3
)
d) Substitute the known values
x1 , y1 , xm , and ym into the formula:
e) Solve the first half of the formula for
the unknown x-coordinate x2 :
g) So, what are the coordinates of the
h) Check it: show that the midpoint of the line segment with one point
B (4,  10) and the other endpoint from part (g) really does have its midpoint at
other endpoint of AB ? Answer as an
ordered pair:
f) Solve the second half of the formula
for the unknown y-coordinate y2 :
M (5, 3) :
6. Suppose we know that one endpoint of CD is located at point C (7, 9) and its midpoint is located at M (3,4) . What are the
coordinates of point D? Make sure you answer as an ordered pair, and you show a check:
Coordinates of point D:
Check:
7. Make up equations for 3 lines that all have the same slope, but different y-intercepts:
Equation 1:
Equation 2:
Equation 3:
8. Make up equations for 3 lines that all have the different slopes, but the same y-intercept:
Equation 4:
Equation 5:
Equation 6:
9. Simplify each of the following:
a)
18
b)
Note: it's not
already
simplified!
e)
36  9
6 5  2 25
c)
73 5
d)
2 5  4 15
The answer
f) Distribute: 2 x( y  z )
g) Simplify: 2 5 (3 
5)
8 75 is not simplified!
h) Simplify: 2 2 (3 10 
4)
10. Solve and check each of the following systems by elimination. Remember that you'll have to set up a "fair fight" to eliminate
one of the variables:
a)
x  y  71
b)
x  y  37
2 x  3 y  26
5 x  4 y  23
Solution: (answer as an ordered pair):
Solution: (answer as an ordered pair):
Check1:
Check1:
Check2:
11. Determine the x-intercept of the graph of the equation
Check2:
3x  5 y 2  21 . Answer as an ordered pair.