1 distance formula standard 5 the distance and midpoint formulas problem 1 problem 2 problem 3...
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1
DISTANCE FORMULA
Standard 5
THE DISTANCE AND MIDPOINT FORMULAS
PROBLEM 1
PROBLEM 2
PROBLEM 3
PROBLEM 4
PROBLEM 5
MIDPOINT FORMULA
END SHOW
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2
Standard 5
Students demonstrate knowledge of how real and complex numbers are related both arithmetically and graphically. In particular, they can plot complex numbers as points in the plane.
Estándar 5:
Los estudiantes demuestran conocimiento de cómo números complejos y reales se relacionan, tanto aritméticamente como geométricamente. En particular pueden graficar números complejos como puntos en el plano.
ALGEBRA II STANDARDS THIS LESSON AIMS:
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3
Distance Formula between two points in a plane:
d = (x –x ) + (y –y )2 2
1 12 2
Standard 5
42 6-2-4-6
2
4
6
-2
-4
-6
8 10-8-10
8
-8
10
x
y
y1x
1
y2x2
,
,
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4
Standard 5
Find the distance between points at A(2, 1) and B(6,4).y1
y2 x1
x2
AB= ( - ) + ( - )2 2
AB= ( -4 ) + ( -3 )2 2
= 16 + 9
= 25
AB=5
2 6 1 4
1
2
3
4
5
6
7
8
9
21 3 4 5 76 8 9 10 x
y
B
A
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5
d = ( - ) + ( - )2 2
= ( 5 ) + ( -1 )2 2
= 25 + 1
2 -3 -6 -5
d = (x –x ) + (y –y )2 2
1 12 2
y1x
1
y2x2
=(-3,-5)
=(2,-6)
d= 26
Standards 5
= ( + ) + ( + )2 22 3 -6 5
42 6-2-4-6
2
4
6
-2
-4
-6
8 10-8-10
8
-8
10
x
y
Find the distance between (-3,-5) and (2,-6).
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6
Standard 5
Find the value of a, so that the distance between (-6,2) and (a,10) be 10 units.
We use the distance formula:
d = (x –x ) + (y –y )2 2
1 12 2
10 = ( - ) + ( - )2 210
y1
2
y2
a
x1
-6
x2
10 = (-6-a) + (-8)22
100 = (-6-a) + 642
22
-64 -64
36 = (-6-a)2
6 = |-6-a|
6 = -(-6-a) 6 = -6-a
6 = 6 + a
-6 -6
a = 0
+6 +6
a = -12
6 = |-6-a|
Check:
6 =|-6- ( )| 6 =|-6- ( )|0 -12
6 = |-6|
6 = 6
6 =|-6+12|
6 =|6|6 = 6
6 = |-6-a|
Solving this absolute value equation:
12 = -a(-1) (-1)
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7
Standard 5
Midpoint of a Line Segment:
If a line segment has endpoints at and , then the midpoint of the line segment has coordinates:
y1x1 y2x2
yx, =x1 x2 ,
2+ y1 y2
2+
21 3-1-2-3
1
2
3
-1
-2
-3
4 5-4-5
4
-4
5
x
y
y1x
1,
y2x2 ,
(x,y)
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8
Find the midpoint of the line segment that connects points (1,6) and (9,8). Show it graphically.
1
2
3
4
5
6
7
8
9
21 3 4 5 76 8 9 10 x
y
(1,6)
(9,8)(5,7)
yx, = ,2+
2+
x1
1
x2
9
y1
6
y2
8
=yx, 142
102
,
=yx, 75,
Standard 5
yx, =x1 x2 ,
2+ y1 y2
2+Using:
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9
Standard 5
yx, =x1 x2 ,
2+ y1 y2
2+
= ,2+
2+
x1
-2
y1
-65
y
8,
x,
Using the Midpoint Formula:
x2y2
5=2+-6 y28=
2+-2 x
2(2) (2) (2) (2)
10 =-6 + y216 =-2 + x2
+2 +2 +6 +6
x2 =18 y2 =16
= 1618,y2x2 ,
y
84 12-4-8-12
4
8
12
-4
-8
-12
16 20-16-20
16
-16
20
xK
M
L
Given the coordinates of one endpoint of KL are K(-2,-6) and its midpoint M(8, 5). What are the coordinates of the other endpoint L. Graph them.
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