1.3: Use Midpoint and Distance Formulas

1.3: Use Midpoint and Distance Formulas
Objectives:
1. To construct a
segment bisector
with a compass and
straightedge
2. To define midpoint
and segment
bisector
3. To use the Midpoint
and Distance
Formulas
Assignment:
• P. 19-22: 2, 4, 6, 10,
14-30 even, 23, 31,
32, 41, 42, 46, 47,
54-57
• Challenge Problems
Objective 1
You will be able to construct a segment
bisector with a compass and straightedge
Perpendicular Bisector
1. Draw a segment.
Label the
endpoints A and B.
Perpendicular Bisector
2. Using the same
compass setting,
draw two intersecting
arcs through the
segment, one
centered at A, the
other centered at B.
Label the
intersection points C
and D.
Perpendicular Bisector
3. Draw a line
through points C
and D.
Perpendicular Bisector
4. Label the new
point of
intersection M. Is
point is called the
midpoint.
Objective 2
You will be able to define
midpoint and segment bisector
Midpoint
The midpoint of a segment is the point on
the segment that divides, or bisects, it into
two congruent segments.
Segment Bisector
A segment bisector is a point, ray, line, line
segment, or plane that intersects the segment
at its midpoint.
Exercise 1
Find 𝐷𝑀 if 𝑀 is the midpoint of 𝐷𝐴, 𝐷𝑀 =
4𝑥 − 1, and 𝑀𝐴 = 3𝑥 + 3.
A
M
D
Exercise 2: SAT
In the figure shown, 𝐴𝐵𝐶𝐷 is
a rectangle with 𝐵𝐶 = 4
and 𝐴𝐵 = 6. Points 𝑃, 𝑄,
and 𝑅 are different points
on a line (not shown) that
is parallel to 𝐴𝐷. Points 𝑃
and 𝑄 are symmetric about
𝐴𝐵 and points 𝑄 and 𝑅 are
symmetric about 𝐶𝐷. What
is 𝑃𝑅?
B
C
P
R
Q
A
D
Note: Figure not drawn to scale.
Exercise 3
𝑂𝑃 lies on a real number line with point 𝑂 at
− 9 and point 𝑃 at 3. Where is the
midpoint of the segment?
O
- 10
P
-5
What if the endpoints of 𝑂𝑃 were at 𝑥1 and
𝑥2 ?
5
You will be able to use the
Midpoint and Distance
Formulas
In the Coordinate Plane
ts on the
ate Plane
A: (-6.00, -2.00)
B: (6.00, 4.00)
4
B
Midpoint: (0.00, 1.00)
2
Midpoint
-5
5
-2
A
We could extend the
previous exercise by
putting the segment in
the coordinate plane.
Now we have two
dimensions and two
sets of coordinates.
Each of these would
have to be averaged
to find the coordinates
of the midpoint.
The Midpoint Formula
If 𝐴 𝑥1 , 𝑦1 and
𝐵 𝑥2 , 𝑦2 are points in
a coordinate plane,
then the midpoint 𝑀 of
𝐴𝐵 has coordinates
𝑥1 + 𝑥2 𝑦1 + 𝑦2
,
2
2
The Midpoint Formula
The coordinates of
the midpoint of a
segment are
basically the
averages of the 𝑥and 𝑦-coordinates
of the endpoints
Exercise 4
Find the midpoint of the segment with
endpoints at (-1, 5) and (3, 8).
Exercise 5
The midpoint 𝐶 of 𝐼𝑁 has coordinates (4, -3).
Find the coordinates of point 𝐼 if point 𝑁 is
at (10, 2).
Exercise 6
Use the Midpoint Formula multiple times to
find the coordinates of the points that
divide 𝐴𝐵 into four congruent segments.
A
B
Parts of a Right Triangle
Which segment is the longest in any right
triangle?
The Pythagorean Theorem
In a right triangle, if 𝑎 and 𝑏
are the lengths of the legs
and 𝑐 is the length of the
hypotenuse, then
𝑐 2 = 𝑎2 + 𝑏 2 .
Exercise 7
How high up on the
wall will a twentyfoot ladder reach if
the foot of the
ladder is placed five
feet from the wall?
The Distance Formula
Sometimes instead of finding a segment’s
midpoint, you want to find it’s length. Notice
how every non-vertical or non-horizontal
segment in the coordinate plane can be
turned into the hypotenuse of a right triangle.
Example 8
Graph 𝐴𝐵 with 𝐴(2, 1) and 𝐵(7, 8). Add
segments to your drawing to create right
∆𝐴𝐵𝐶. Now use the Pythagorean Theorem
to find 𝐴𝐵.
Distance Formula
In the previous problem, you found the
length of a segment by connecting it to a
right triangle on graph paper and then
applying the Pythagorean Theorem. But
what if the points are too far apart to be
conveniently graphed on a piece of
ordinary graph paper? For example, what
is the distance between the points (15, 37)
and (42, 73)? What we need is a formula!
The Distance Formula
To find the distance between
points 𝐴 and 𝐵 shown at the
right, you can simply count
the squares on the side 𝐴𝐶
and the squares on side 𝐵𝐶,
then use the Pythagorean
Theorem to find 𝐴𝐵. But if
the distances are too great to
count conveniently, there is a
simple way to find the
lengths. Just use the Ruler
Postulate.
B
8
6
4
2
A
C
5
The Distance Formula
You can find the horizontal
distance subtracting the 𝑥coordinates of points 𝐴 and
𝐵: 𝐴𝐶 = |7 – 2| = 5. Similarly,
to find the vertical distance
𝐵𝐶, subtract the 𝑦coordinates of points 𝐴 and
𝐵: 𝐵𝐶 = |8 – 1| = 7. Now you
can use the Pythagorean
Theorem to find 𝐴𝐵.
B
8
6
4
2
A
C
5
Exercise 9
Generalize this result
and come up with a
formula for the
distance between
any two points
(x1, y1) and (x2, y2).
B
8
(x 2, y 2)
6
4
2
(x 1, y 1)
(x 2, y 1)
A
C
5
The Distance Formula
If the coordinates of points 𝐴 and 𝐵 are 𝑥1 , 𝑦1
and 𝑥2 , 𝑦2 , then 𝐴𝐵 = 𝑥2 − 𝑥1 2 + 𝑦2 − 𝑦1
2
Exercise 10
To the nearest tenth of a unit, what is the
approximate length of 𝑅𝑆, with endpoints
𝑅(3, 1) and 𝑆(−1, −5)?
Exercise 11
A coordinate grid is placed over a map. City
𝐴 is located at (-3, 2) and City 𝐵 is located
at (4, 8). If City 𝐶 is at the midpoint
between City 𝐴 and City 𝐵, what is the
approximate distance in coordinate units
from City 𝐴 to City 𝐶?
Exercise 12
Points on a 3-Dimensional
coordinate grid can be
located with coordinates of
the form (x, y, z). Finding
the midpoint of a segment
or the length of a segment
in 3-D is analogous to
finding them in 2-D, you
just have 3 coordinates
with which to work.
Exercise 12
Find the midpoint and
the length of the
segment with
endpoints (2, 5, 8) and
(-3, 1, 2).
1.3: Use Midpoint and Distance Formulas
Objectives:
1. To construct a
segment bisector
with a compass and
straightedge
2. To define midpoint
and segment
bisector
3. To use the Midpoint
and Distance
Formulas
Assignment:
• P. 19-22: 2, 4, 6, 10,
14-30 even, 23, 31,
32, 41, 42, 46, 47,
54-57
• Challenge Problems