geometry - mesa public schools · midpoint 1.3 using midpoint and distance formulas august 20, 2015...
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Geometry1.3 Using Midpoint and Distance Formulas
August 20, 20151.3 Using Midpoint and Distance Formulas
1.3 Essential Question:
• How is the segment length found using the midpoint
and segment bisector and how are the midpoint and
distance formulas used?
August 20, 20151.3 Using Midpoint and Distance Formulas
Midpoint
August 20, 20151.3 Using Midpoint and Distance Formulas
A M C
•M is the midpoint of segment AC.
•M bisects segment AC.
•Bisect: divide into two congruent parts.
•Congruence marks indicate congruent segments.
MCAM
August 20, 20151.3 Using Midpoint and Distance Formulas
A Segment Bisector can be a…
Segment Ray
Line Plane
Example 1
•
August 20, 20151.3 Using Midpoint and Distance Formulas
Example 2
•
August 20, 20151.3 Using Midpoint and Distance Formulas
Your Turn
Find EF.
August 20, 20151.3 Using Midpoint and Distance Formulas
August 20, 20151.3 Using Midpoint and Distance Formulas
The Coordinate Plane
x-axis
y-axis
Points are located
with a coordinate pair,
(x, y). 3
4
(3, 4)
-6
2
(-6, 2)
August 20, 20151.3 Using Midpoint and Distance Formulas
Midpoint Formula
(x1, y1)
(x2, y2)
M
2,
2
2121 yyxx
The midpoint is
the average point.
August 20, 20151.3 Using Midpoint and Distance Formulas
Example 3
Find the midpoint of segment AB if A is point (10, -6) and B is
point (2, 8).
Solution:
2
2,
2
12
2
86,
2
210(6, 1)
August 20, 20151.3 Using Midpoint and Distance Formulas
Your Turn
Find the midpoint of the segment between (20, -14) and
(-16, 4).
Solution:
2
10,
2
4
2
414,
2
)16(20(2, -5)
August 20, 20151.3 Using Midpoint and Distance Formulas
Example 4
The midpoint of segment DH is O(3, 4). One endpoint is D(5, 7).
Find the coordinates of H.
D(5,7)
O(3,4)
H(x, y)
Write out the formula:
2
7,
2
5 yx
continues…
August 20, 20151.3 Using Midpoint and Distance Formulas
Example 4 Continued
There are two equations to solve:
Solve:
1
65
32
5
x
x
x Solve:
1
87
42
7
y
y
y
2
7,
2
5 yx 4,3
continues…
August 20, 20151.3 Using Midpoint and Distance Formulas
Example 4 Continued
Since x = 1 and y = 1, the other endpoint is H(1,1).
D(5,7)
O(3,4)
H(1, 1)
August 20, 20151.3 Using Midpoint and Distance Formulas
Your Turn
The midpoint of segment AB is M(1, 5). If one endpoint is
A(-3, -4), find the coordinates of B.
Solution:
2
)4(,
2
)3( yx
Solve for x:
5
23
12
3
x
x
x Solve for y:
14
104
52
4
y
y
y
Endpoint: B(5, 14)
5,1
August 20, 20151.3 Using Midpoint and Distance Formulas
Radical Review
7 Can this be simplified? NO!
12 What about this one? YES!
3212
This is called Simplifying a Radical.4 3
2 2
August 20, 20151.3 Using Midpoint and Distance Formulas
Let’s Practice
2027 32
August 20, 20151.3 Using Midpoint and Distance Formulas
Try it. Simplify these radicals:
8200
5044
a.
d.c.
b.
August 20, 20151.3 Using Midpoint and Distance Formulas
Put down your pencils, watch, listen, and think!
August 20, 20151.3 Using Midpoint and Distance Formulas
Distances
x-axis
y-axis
(5, 3)
How far is (5, 3) from
(0, 0)?
?
5
3
This is a right
triangle.
August 20, 20151.3 Using Midpoint and Distance Formulas
The Pythagorean Theorem
a
b
c
a2 + b2 = c2
A(x1, y1) C(x2, y1)
B(x2, y2)
dy2 – y1
x2 – x1
2a2
2 1( - )x x
2 22 1 2 1( - ) + ( - ) =x x y y d
Rather than thinking through the Pythagorean Theorem to find
the distance between two points, we can use the following
reasoning to obtain the distance formula.
2+ b 2=c2
2 1+ ( - )y y 2=d
August 20, 20151.3 Using Midpoint and Distance Formulas
Distance Formula
The distance d between two points (x1, y1) and
(x2, y2) is:
2 22 1 2 1= ( - ) + ( - )d x x y y
August 20, 20151.3 Using Midpoint and Distance Formulas
August 20, 20151.3 Using Midpoint and Distance Formulas
Example 5
Find the distance between (6, 8) and (3, 4).
212
2
12 yyxxd
Does it matter which
point is chosen as
point 1 or point 2?
5
25
169
43
4836
22
22
d
x1 y1 x2 y2
August 20, 20151.3 Using Midpoint and Distance Formulas
Example 6
Find the distance between (-6, 2) and (0, 4).
102
10440
436
26
24)6(0
22
22
d
212
2
12 yyxxd
x1 y1 x2 y2
August 20, 20151.3 Using Midpoint and Distance Formulas
Your Turn
Find the distance between (9, 10) and (6, 9).
10
19
13
91069
22
22
d
x1 y1 x2 y2
212
2
12 yyxxd
August 20, 20151.3 Using Midpoint and Distance Formulas
Another Problem
172
17468
464
)2()8(
)9(735
22
22
d
Find the distance between (-5, -7) and (3, -9).x1 y1 x2 y2
212
2
12 yyxxd
Assignment
August 20, 20151.3 Using Midpoint and Distance Formulas
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