Geometry - Mesa Public Schools · Midpoint 1.3 Using Midpoint and Distance Formulas August 20, 2015...
Transcript of Geometry - Mesa Public Schools · Midpoint 1.3 Using Midpoint and Distance Formulas August 20, 2015...
![Page 1: Geometry - Mesa Public Schools · Midpoint 1.3 Using Midpoint and Distance Formulas August 20, 2015 A M C •M is the midpoint of segment AC. •M bisects segment AC. •Bisect: divide](https://reader031.fdocuments.in/reader031/viewer/2022022511/5aded6097f8b9af05b8ba027/html5/thumbnails/1.jpg)
Geometry1.3 Using Midpoint and Distance Formulas
![Page 2: Geometry - Mesa Public Schools · Midpoint 1.3 Using Midpoint and Distance Formulas August 20, 2015 A M C •M is the midpoint of segment AC. •M bisects segment AC. •Bisect: divide](https://reader031.fdocuments.in/reader031/viewer/2022022511/5aded6097f8b9af05b8ba027/html5/thumbnails/2.jpg)
August 20, 20151.3 Using Midpoint and Distance Formulas
![Page 3: Geometry - Mesa Public Schools · Midpoint 1.3 Using Midpoint and Distance Formulas August 20, 2015 A M C •M is the midpoint of segment AC. •M bisects segment AC. •Bisect: divide](https://reader031.fdocuments.in/reader031/viewer/2022022511/5aded6097f8b9af05b8ba027/html5/thumbnails/3.jpg)
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
![Page 4: Geometry - Mesa Public Schools · Midpoint 1.3 Using Midpoint and Distance Formulas August 20, 2015 A M C •M is the midpoint of segment AC. •M bisects segment AC. •Bisect: divide](https://reader031.fdocuments.in/reader031/viewer/2022022511/5aded6097f8b9af05b8ba027/html5/thumbnails/4.jpg)
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
![Page 5: Geometry - Mesa Public Schools · Midpoint 1.3 Using Midpoint and Distance Formulas August 20, 2015 A M C •M is the midpoint of segment AC. •M bisects segment AC. •Bisect: divide](https://reader031.fdocuments.in/reader031/viewer/2022022511/5aded6097f8b9af05b8ba027/html5/thumbnails/5.jpg)
August 20, 20151.3 Using Midpoint and Distance Formulas
A Segment Bisector can be a…
Segment Ray
Line Plane
![Page 6: Geometry - Mesa Public Schools · Midpoint 1.3 Using Midpoint and Distance Formulas August 20, 2015 A M C •M is the midpoint of segment AC. •M bisects segment AC. •Bisect: divide](https://reader031.fdocuments.in/reader031/viewer/2022022511/5aded6097f8b9af05b8ba027/html5/thumbnails/6.jpg)
Example 1
•
August 20, 20151.3 Using Midpoint and Distance Formulas
![Page 7: Geometry - Mesa Public Schools · Midpoint 1.3 Using Midpoint and Distance Formulas August 20, 2015 A M C •M is the midpoint of segment AC. •M bisects segment AC. •Bisect: divide](https://reader031.fdocuments.in/reader031/viewer/2022022511/5aded6097f8b9af05b8ba027/html5/thumbnails/7.jpg)
Example 2
•
August 20, 20151.3 Using Midpoint and Distance Formulas
![Page 8: Geometry - Mesa Public Schools · Midpoint 1.3 Using Midpoint and Distance Formulas August 20, 2015 A M C •M is the midpoint of segment AC. •M bisects segment AC. •Bisect: divide](https://reader031.fdocuments.in/reader031/viewer/2022022511/5aded6097f8b9af05b8ba027/html5/thumbnails/8.jpg)
Your Turn
Find EF.
August 20, 20151.3 Using Midpoint and Distance Formulas
![Page 9: Geometry - Mesa Public Schools · Midpoint 1.3 Using Midpoint and Distance Formulas August 20, 2015 A M C •M is the midpoint of segment AC. •M bisects segment AC. •Bisect: divide](https://reader031.fdocuments.in/reader031/viewer/2022022511/5aded6097f8b9af05b8ba027/html5/thumbnails/9.jpg)
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)
![Page 10: Geometry - Mesa Public Schools · Midpoint 1.3 Using Midpoint and Distance Formulas August 20, 2015 A M C •M is the midpoint of segment AC. •M bisects segment AC. •Bisect: divide](https://reader031.fdocuments.in/reader031/viewer/2022022511/5aded6097f8b9af05b8ba027/html5/thumbnails/10.jpg)
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.
![Page 11: Geometry - Mesa Public Schools · Midpoint 1.3 Using Midpoint and Distance Formulas August 20, 2015 A M C •M is the midpoint of segment AC. •M bisects segment AC. •Bisect: divide](https://reader031.fdocuments.in/reader031/viewer/2022022511/5aded6097f8b9af05b8ba027/html5/thumbnails/11.jpg)
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)
![Page 12: Geometry - Mesa Public Schools · Midpoint 1.3 Using Midpoint and Distance Formulas August 20, 2015 A M C •M is the midpoint of segment AC. •M bisects segment AC. •Bisect: divide](https://reader031.fdocuments.in/reader031/viewer/2022022511/5aded6097f8b9af05b8ba027/html5/thumbnails/12.jpg)
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)
![Page 13: Geometry - Mesa Public Schools · Midpoint 1.3 Using Midpoint and Distance Formulas August 20, 2015 A M C •M is the midpoint of segment AC. •M bisects segment AC. •Bisect: divide](https://reader031.fdocuments.in/reader031/viewer/2022022511/5aded6097f8b9af05b8ba027/html5/thumbnails/13.jpg)
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…
![Page 14: Geometry - Mesa Public Schools · Midpoint 1.3 Using Midpoint and Distance Formulas August 20, 2015 A M C •M is the midpoint of segment AC. •M bisects segment AC. •Bisect: divide](https://reader031.fdocuments.in/reader031/viewer/2022022511/5aded6097f8b9af05b8ba027/html5/thumbnails/14.jpg)
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…
![Page 15: Geometry - Mesa Public Schools · Midpoint 1.3 Using Midpoint and Distance Formulas August 20, 2015 A M C •M is the midpoint of segment AC. •M bisects segment AC. •Bisect: divide](https://reader031.fdocuments.in/reader031/viewer/2022022511/5aded6097f8b9af05b8ba027/html5/thumbnails/15.jpg)
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)
![Page 16: Geometry - Mesa Public Schools · Midpoint 1.3 Using Midpoint and Distance Formulas August 20, 2015 A M C •M is the midpoint of segment AC. •M bisects segment AC. •Bisect: divide](https://reader031.fdocuments.in/reader031/viewer/2022022511/5aded6097f8b9af05b8ba027/html5/thumbnails/16.jpg)
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
![Page 17: Geometry - Mesa Public Schools · Midpoint 1.3 Using Midpoint and Distance Formulas August 20, 2015 A M C •M is the midpoint of segment AC. •M bisects segment AC. •Bisect: divide](https://reader031.fdocuments.in/reader031/viewer/2022022511/5aded6097f8b9af05b8ba027/html5/thumbnails/17.jpg)
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
![Page 18: Geometry - Mesa Public Schools · Midpoint 1.3 Using Midpoint and Distance Formulas August 20, 2015 A M C •M is the midpoint of segment AC. •M bisects segment AC. •Bisect: divide](https://reader031.fdocuments.in/reader031/viewer/2022022511/5aded6097f8b9af05b8ba027/html5/thumbnails/18.jpg)
August 20, 20151.3 Using Midpoint and Distance Formulas
Let’s Practice
2027 32
![Page 19: Geometry - Mesa Public Schools · Midpoint 1.3 Using Midpoint and Distance Formulas August 20, 2015 A M C •M is the midpoint of segment AC. •M bisects segment AC. •Bisect: divide](https://reader031.fdocuments.in/reader031/viewer/2022022511/5aded6097f8b9af05b8ba027/html5/thumbnails/19.jpg)
August 20, 20151.3 Using Midpoint and Distance Formulas
Try it. Simplify these radicals:
8200
5044
a.
d.c.
b.
![Page 20: Geometry - Mesa Public Schools · Midpoint 1.3 Using Midpoint and Distance Formulas August 20, 2015 A M C •M is the midpoint of segment AC. •M bisects segment AC. •Bisect: divide](https://reader031.fdocuments.in/reader031/viewer/2022022511/5aded6097f8b9af05b8ba027/html5/thumbnails/20.jpg)
August 20, 20151.3 Using Midpoint and Distance Formulas
Put down your pencils, watch, listen, and think!
![Page 21: Geometry - Mesa Public Schools · Midpoint 1.3 Using Midpoint and Distance Formulas August 20, 2015 A M C •M is the midpoint of segment AC. •M bisects segment AC. •Bisect: divide](https://reader031.fdocuments.in/reader031/viewer/2022022511/5aded6097f8b9af05b8ba027/html5/thumbnails/21.jpg)
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.
![Page 22: Geometry - Mesa Public Schools · Midpoint 1.3 Using Midpoint and Distance Formulas August 20, 2015 A M C •M is the midpoint of segment AC. •M bisects segment AC. •Bisect: divide](https://reader031.fdocuments.in/reader031/viewer/2022022511/5aded6097f8b9af05b8ba027/html5/thumbnails/22.jpg)
August 20, 20151.3 Using Midpoint and Distance Formulas
The Pythagorean Theorem
a
b
c
a2 + b2 = c2
![Page 23: Geometry - Mesa Public Schools · Midpoint 1.3 Using Midpoint and Distance Formulas August 20, 2015 A M C •M is the midpoint of segment AC. •M bisects segment AC. •Bisect: divide](https://reader031.fdocuments.in/reader031/viewer/2022022511/5aded6097f8b9af05b8ba027/html5/thumbnails/23.jpg)
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
![Page 24: Geometry - Mesa Public Schools · Midpoint 1.3 Using Midpoint and Distance Formulas August 20, 2015 A M C •M is the midpoint of segment AC. •M bisects segment AC. •Bisect: divide](https://reader031.fdocuments.in/reader031/viewer/2022022511/5aded6097f8b9af05b8ba027/html5/thumbnails/24.jpg)
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
![Page 25: Geometry - Mesa Public Schools · Midpoint 1.3 Using Midpoint and Distance Formulas August 20, 2015 A M C •M is the midpoint of segment AC. •M bisects segment AC. •Bisect: divide](https://reader031.fdocuments.in/reader031/viewer/2022022511/5aded6097f8b9af05b8ba027/html5/thumbnails/25.jpg)
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
![Page 26: Geometry - Mesa Public Schools · Midpoint 1.3 Using Midpoint and Distance Formulas August 20, 2015 A M C •M is the midpoint of segment AC. •M bisects segment AC. •Bisect: divide](https://reader031.fdocuments.in/reader031/viewer/2022022511/5aded6097f8b9af05b8ba027/html5/thumbnails/26.jpg)
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
![Page 27: Geometry - Mesa Public Schools · Midpoint 1.3 Using Midpoint and Distance Formulas August 20, 2015 A M C •M is the midpoint of segment AC. •M bisects segment AC. •Bisect: divide](https://reader031.fdocuments.in/reader031/viewer/2022022511/5aded6097f8b9af05b8ba027/html5/thumbnails/27.jpg)
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
![Page 28: Geometry - Mesa Public Schools · Midpoint 1.3 Using Midpoint and Distance Formulas August 20, 2015 A M C •M is the midpoint of segment AC. •M bisects segment AC. •Bisect: divide](https://reader031.fdocuments.in/reader031/viewer/2022022511/5aded6097f8b9af05b8ba027/html5/thumbnails/28.jpg)
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
![Page 29: Geometry - Mesa Public Schools · Midpoint 1.3 Using Midpoint and Distance Formulas August 20, 2015 A M C •M is the midpoint of segment AC. •M bisects segment AC. •Bisect: divide](https://reader031.fdocuments.in/reader031/viewer/2022022511/5aded6097f8b9af05b8ba027/html5/thumbnails/29.jpg)
Assignment
August 20, 20151.3 Using Midpoint and Distance Formulas