1. SEGMENT ADDITION POSTULATE
LINES
PLANES
STANDARDS 13, 17
SEGMENT ADDITION POSTULATE
PROBLEM 1A
PROBLEM 1B
MIDPOINT
PROBLEM 2A
PROBLEM 2B
COORDINATE GEOMETRY
PROBLEM 3A
PROBLEM 3B
DISTANCE FORMULA
PROBLEM 4
PROBLEM 5
PROBLEM 6
PYTHAGOREAN THEOREM
PROBLEM 7
PROBLEM 8
MIDPOINT FORMULA
PROBLEM 9A
PROBLEM 9B
PROBLEM 10A
PROBLEM 10B
END SHOW
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2. STANDARD 13:
Students prove relationships between angles in polygons using properties of complementary, supplementary, vertical and exterior angles.
STANDARD 15:
Students use the Pythagorean Theorem to determine distance and find missing lengths of sides of right triangles.
STANDARD 17:
Students prove theorems by using coordinate geometry, including the midpoint of a line segment, the distance formula, and various forms of equations of lines and circles.
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3. ESTÁNDAR 13:
Los estudiantes prueban relaciones entre ángulos en polígonos usando propiedades de ángulos complementarios, suplementarios, verticales y ángulos exteriores.
ESTÁNDAR 15:
Los estudiantes usan el Teorema de Pitágoras para determinar distancias y encontrar las longitudes de los lados de triángulos.
ESTÁNDAR 17:
Los estudiantes prueban teoremas usando geometría coordenada, incluyendo el punto medio de un segmento, la fórmula de la distancia y varias formas de ecuaciones de líneas y círculos.
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4. LINES STANDARD 13 Line AB B p A or Line p How do we call this line? H
Line HI
is VERTICAL
What about this other? M N
Line MN
is HORIZONTAL
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5. LINES STANDARD 13 F E m D C How many different ways can we call
Line m?
Line CD
Line DC
Line CE
Line EC OR
Line CF
Line FC
Line DE
Can you figure out other names?
Line DF
Line EF
IN GENERAL A LINE IS NAMED BY TWO POINTS.
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6. SEGMENTS STANDARD 13 If we have line DE
and we take one part of the line D E
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7. SEGMENTS STANDARD 13 If we have line DE
and we take one part of the line D E D E
then this part is called:
LINE SEGMENT DE
Can you name the different line segments in the following line: S T U ST TS
SEGMENTS: OR SU US TU UT
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8. RAYS STANDARD 13 B A This is RAY AB
What are the differences between a Line, a Line Segment and a Ray? M N
The line is infinite and never ends at either side. D E
The line segment has two endpoints.
The ray has on one side one endpoint at the other side it never ends it goes on to infinite. A B
What do they have in common?
They are named using TWO POINTS.
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9. COLLINEAR VS NONCOLLINEAR
STANDARD 13
COLLINEAR VS NONCOLLINEAR
Points C, D, E and F: are they COLLINEAR? C F D E
Yes, they are COLLINEAR because they lie in the same line
Are points A, B, C, and D collinear? A C B D
No, they are NONCOLLINEAR because they don’t lie in the same line.
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10. STANDARD 13
Can you explain where the following two lines intersect? l m A B C D E
They INTERSECT at point E. NO
Can they intersect at other point?
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11. PLANES STANDARD 13 A M This is: D B PLANE M PLANE ADC C PLANE BDC
PLANE ABD
In general a Plane can be named using three non-collinear points.
The figure above represents a PLANE, which is a flat surface that has no end at any of the sides.
What examples can you give of objects lying in a PLANE?
The wall of a house
The lid of a shoebox.
The ground of a football field.
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12. STANDARD 13 P M A B l m n
Can you describe the INTERSECTION of planes M and P?
Planes M and P intersect at line AB.
PLANES ALWAYS INTERSECT AT A LINE.
Where do lines m and l intersect?
They intersect at point B. LINES ALWAYS INTERSECT AT A POINT.
Where do lines n and l intersect?
At point A.
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13. P Q STANDARD 13 A L D K M B H C Are Points L, K, and M COPLANAR?
Yes, they are COPLANAR because they LIE ON THE SAME PLANE P.
Is point H, coplanar with points L, K, and M?
No, because it lies on plane Q and points L, K, and M are in different plane, on plane P.
NON-COPLANAR points are points that lie in different planes.
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14. P Q STANDARD 13 A D B C On what planes does point C lie?
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15. P Q STANDARD 13 A D B C On what planes does point C lies?
On planes P and Q.
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16. P Q STANDARD 13 A D B C On what planes does point D lie?
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17. P Q STANDARD 13 A D B C On what planes does point D lie?
It only lies on plane Q.
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18. P Q STANDARD 13 A D B C k On what plane is line k lying?
Since points B and D lie on plane Q
then line k lies on its entirety on plane Q
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19. SUMMARIZING FINDINGS:
STANDARD 13
SUMMARIZING FINDINGS:
Through any two points there is exactly one line.
Through any three points not on the same line there is exactly one plane or through any three points non-collinear there is one plane.
A line contains at least two points.
A plane contains at least three points not on the same line.
A plane contains at least three non-collinear points.
If two points lie in a plane, the entire line containing those points lies in that plane.
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20. SEGMENT ADDITION POSTULATE. If B is between A and C then AB + BC = AC.
STANDARD 17 A B C AC AB BC + =
SEGMENT ADDITION POSTULATE.
If B is between A and C then AB + BC = AC.
If AB + BC = AC, then B is between A and C.
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21. STANDARD 17
Find the length for AB and BC if AC = 60 and AB = 4x + 6 and BC= 6x (B is between A and C) AC AB BC
Applying Segment Addition Postulate: A B C + =
Now finding AB and BC: + = AB BC AC
4x +6
+ 6x + 14
= 60
AB = 4x + 6
BC = 6x + 14
4x + 6x = 60
= 4( ) + 6 4
= 6( ) + 14 4
10x + 20 = 60 = =
= 22
= 38
10x = 40
Verifying the solution:
x = 4
= 60
60 = 60
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22. STANDARD 17
Find the length of EF and FG if EG = 80 and EF = 3x + 8 and FG= 7x (F is between E and G) EG EF FG
Applying Segment Addition Postulate: E F G + =
Now finding EF and FG: + = EF FG EG
3x +8
+ 7x + 12
= 80
EF = 3x + 8
FG = 7x + 12
3x + 7x = 80
= 3( ) + 8 6
= 7( ) + 12 6
10x + 20 = 80 = =
= 26
= 54
10x = 60
Verifying the solution:
x = 6
= 80
80 = 80
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23. and we place point B at the same
STANDARD 17
MIDPOINT OF A SEGMENT: A C B
and we place point B at the same
distance from point A than from Point C, then:
If we have segment AC,
Means congruent AB BC
AB = BC
and then point B is THE MIDPOINT OF SEGMENT AC.
Point B is also BISECTING segment AC, because it is dividing it into two halves.
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24. Applying Segment Addition Postulate:
STANDARD 17
AC is bisected by ED. AB= 6X + 8 and BC=4X Find the length for AC. A B C D E AB BC
Now finding AB:
Applying Segment Addition Postulate:
AB = BC
AB = 6X + 8
AC = AB + BC
6X + 8 = 4X + 18
= 6( ) + 8 5
AC = AB + BC
AC = =
6X = 4X + 10
AC = 76
-4X -4X
= 38
2X = 10
Since AB = BC
BC = 38
X = 5
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25. Applying Segment Addition Postulate:
STANDARD 17
RT is bisected by VU. RS= 8X + 4 and ST=4X Find the length for RT. R S T V U RS ST
Now finding RS:
Applying Segment Addition Postulate:
RS = ST
RS = 8X + 4
RT = RS + ST
8X + 4 = 4X + 28
= 8( ) + 4 6
RT = RS + ST
RT = =
8X = 4X + 24
RT = 104
-4X -4X
= 52
4X = 24
Since RS = ST
ST = 52
X = 6
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26. CARTESIAN COORDINATE PLANE
STANDARD 17
CARTESIAN COORDINATE PLANE
y-axis 10
Quadrant II 8
Quadrant I 6 4
Origin 2
-10 -8 -6 -4 -2 O 2 4 6 8 10
x-axis -2 -4
Quadrant III
Quadrant IV -6 -8
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27. CARTESIAN COORDINATE PLANE
STANDARD 17
CARTESIAN COORDINATE PLANE
y-axis 10 8
y-coordinate 6 4
(9, 4) 2
x-coordinate
-10 -8 -6 -4 -2 O 2 4 6 8 10
x-axis -2 -4 -6 -8
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28. CARTESIAN COORDINATE PLANE
STANDARD 17
CARTESIAN COORDINATE PLANE
y-axis 10 8 6 4 2
-10 -8 -6 -4 -2 O 2 4 6 8 10
x-axis -2 -4
y-coordinate -6 -8
(10,-8)
x-coordinate
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29. CARTESIAN COORDINATE PLANE
STANDARD 17
CARTESIAN COORDINATE PLANE
y-axis 10 8 6 4 2
y-coordinate
-10 -8 -6 -4 -2 O 2 4 6 8 10
x-axis -2
(-9,-3) -4
x-coordinate -6 -8
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30. DISTANCE FORMULA in a number line is given by:
STANDARD 17
DISTANCE FORMULA in a number line is given by:
|a – b| E D H -6 -4 -2 2 4 6 8 10 12
Find measure of DE, and EH:
DE =
|-2 – (-6)|
EH =
|12 – (-2)|
= |-2 + 6|
= |12 + 2|
= |4|
= |14|
= 4
= 14
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31. DISTANCE FORMULA in a number line is given by:
STANDARD 17
DISTANCE FORMULA in a number line is given by:
|a – b| Q R T -6 -4 -2 2 4 6 8 10 12
Find measure of RT, and QT:
RT =
|12 – (-6)|
QT =
|12 – 6|
= |12 + 6|
= |6|
= |18|
= 6
= 18
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32. Distance Formula between two points in a plane:
STANDARD 17
Distance Formula between two points in a plane:
d = (x –x ) + (y –y ) 2 1 4 2 6 -2 -4 -6 8 10 -8
-10 x y y 1 x , y 2 x ,
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33. Find the distance between points at A(2, 1) and B(6,4).
STANDARD 17
d = (x –x ) + (y –y ) 2 1
Remember: x 2 y 2 x 1 y 1
Find the distance between points at A(2, 1) and B(6,4). 1 2 3 4 5 6 7 8 9 10 x y
AB= ( - ) + ( - ) 2 2 6 1 4
AB= ( -4 ) + ( -3 ) 2 = B
= 25 A
AB=5
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34. Find the distance between (-3,-5) and (2,-6). y x =(-3,-5)
STANDARD 17
Find the distance between (-3,-5) and (2,-6). y 1 x
=(-3,-5) 4 2 6 -2 -4 -6 8 10 -8
-10 x y y 2 x
=(2,-6)
d = (x –x ) + (y –y ) 2 1
d = ( ) + ( ) 2 2 -3 -6 -5
= ( ) + ( ) 2 3 -6 5
= ( 5 ) + ( -1 ) 2 =
d= 26
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35. We use the distance formula: Solving this absolute value equation:
STANDARD 17 -6 x 2 a x 1 2 y 10 y 1
Find the value of a, so that the distance between (-6,2) and (a,10) be 10 units.
We use the distance formula:
Solving this absolute value equation:
d = (x –x ) + (y –y ) 2 1
6 = |-6-a|
10 = ( ) + ( ) 2
6 = -(-6-a)
6 = -6-a
6 = 6 + a 2
10 = (-6-a) + (-8) 2
(-1) (-1)
12 = -a
100 = (-6-a) + 64 2
a = -12
a = 0
Check:
6 = |-6-a|
36 = (-6-a) 2
6 =|-6- ( )|
6 =|-6- ( )|
-12
6 = |-6-a|
6 =|-6+12|
6 = |-6|
6 =|6|
6 = 6
6 = 6
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36. Right triangle parts: STANDARD 15 hypotenuse LEG Right Angle = 90° LEG
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37. Pythagorean Theorem: x y y + z = x z STANDARD 152 z
The square of the hypotenuse is equal to the sum of the square of the legs.
The Pythagorean Theorem applies ONLY to RIGHT TRIANGLES!
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38. Find the value for x: y + z = x x= ? y= 4 4 + 3 = x 16 + 9 = x 25 = x
STANDARD 15
Find the value for x:
y + z = x 2
x= ?
y= 4
= x 2
= x 2
25 = x 2
z= 3
25 = x 2
|x|= 5
x= 5
x= -5
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39. Find the value for z: y= 8 y + z = x z= ? x= 10 8 + z = 10
STANDARD 15
Find the value for z:
y= 8
y + z = x 2
z= ?
x= 10
z = 10 2 2
64 + z = 100
z = 36 2
z = 36 2
|z|= 6
z= 6
z= -6
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40. MIDPOINT FORMULA in a number line is given by:
STANDARD 17
MIDPOINT FORMULA in a number line is given by:
a + b 2 R T -6 -4 -2 2 4 6 8 10 12
Find Midpoint Q of RT: Q 2 = 6 2
= 3
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41. MIDPOINT FORMULA in a number line is given by:
STANDARD 17
MIDPOINT FORMULA in a number line is given by:
a + b 2 K L -4 -2 2 4 6 8 10 12 14
Find Midpoint R of KL: R 2 = 10 2
= 5
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42. Midpoint of a Line Segment:
STANDARD 17
Midpoint of a Line Segment:
If a line segment has endpoints at and , then the midpoint of the line segment has coordinates: y 1 x 2 x 1 2 , + y 1 2 + = x, y 2 1 3 -1 -2 -3 4 5 -4 -5 x y ,
(x,y)
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43. STANDARD 17 x , + y + = Using: x, y x x 9 y 6 y 82 , + y 1 2 + =
Using: x, y x 1 x 2 9 y 1 6 y 2 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 10 x y = , 2 +
(9,8) x, y
(5,7)
(1,6) = y x, 14 2 10 , = y x, 7 5,
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44. STANDARD 17 x , + y + = Using: x, y x 4 x 8 y 7 y 92 , + y 1 2 + =
Using: x, y x 1 4 x 2 8 y 1 7 y 2 9
Find the midpoint of the line segment that connects points (4,7) and (8,9). Show it graphically. 1 2 3 4 5 6 7 8 9 10 x y
(8,9) = , 2 +
(6,8) x, y
(4,7) = y x, 16 2 12 , = y x, 8 6,
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45. Using the Midpoint Formula: L
STANDARD 17 x 1 -2 y 1 -6
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. 8, x, 5 y y 8 4 12 -4 -8
-12 16 20
-16
-20 x
Using the Midpoint Formula: L y x, = x 1 2 , + M = , 2 + x 2 y 2 K 8= 2 + -2 x 5= 2 + -6 y
(2) (2)
(2) (2)
16 =-2 + x 2
10 =-6 + y 2 y 2
=16 x 2
=18 = 16
18, y 2 x ,
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46. Using the Midpoint Formula:
STANDARD 17 x 1 -1 y 1 -5
Given the coordinates of one endpoint of KL are K(-1,-5) and its midpoint M(7, 4). What are the coordinates of the other endpoint L. Graph them. 7, x, 4 y y 8 4 12 -4 -8
-12 16 20
-16
-20 x
Using the Midpoint Formula: y x, = x 1 2 , + L M = , 2 + x 2 y 2 K 7= 2 + -1 x 4= 2 + -5 y
(2) (2)
(2) (2)
14 =-1 + x 2
8 =-5 + y 2 y 2
=13 x 2
=15 = 13
15, y 2 x ,
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