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# Transformations, midpoint, distance - Guided notes

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```​Chapter 6: COORDINATE GEOMETRY - Midpoint
Essential question:​ H
​ ow do you determine the midpoint between two points on a coordinate
plane?
Review:
● Bisect
Sketch CD with midpoint E. Mark drawing
correctly.
● Midpoint
Determine Graphically
Determine Algebraically
Midpoint formula: ​The coordinates for the
midpoint are
x1 +x2 y 1 +y 2
​ ( 2 ,
2 )
Graph and determine midpoint C of a line
segment with endpoints (4, 5) and (-6, -3)
Determine the midpoint D of a line segment
with endpoints (0,-4) and (-5,-4)
Determine the missing endpoint
Determine the missing endpoint
The midpoint of
is (-6, -7).
The coordinates of A
​ ​ are (-5, 8).
Find the coordinates of B
​ ​.
The midpoint of
is (9, 10).
The coordinates of A
​ ​ are (-3, 3).
Find the coordinates of B
​ ​.
1
DISTANCE
Essential question:​ H
​ ow do you determine the distance between two points on a coordinate
plane?
How do you find the distance Find the distance between the endpoints of AB
to endpoints of vertical or
horizontal line segments?
Find the distance between C(3, -4) and D(3, -2)
Find the distance between
R(25, 3) and S(25, 22)
The map of a town is placed on
a coordinate grid. Each unit on
the map represents 1 mile.
Find the distance between the
middle school and the high
school.
On a map, Rachel’s house is
located at (2, 1). Each unit on
the map represents one block.
The dog park is 5 blocks south
of Rachel’s house. Graph
Rachel’s house and the dog park
on the graph?
2
Label the parts of the right triangle: (i.e. a is Pythagorean Theorem:
a leg)
● If a triangle is a right triangle then
_________________________
Solve for x
Distance in Coordinate Geometry
●
Determine the distance between
the two points below
Distance formula
Determine the distance between (length of segment)
the following points:
AB
BC
3
TRANSLATIONS
Essential Question: ​What is the ordered pair rule for a translation?
Translation:​ A translation is a _______________________________ which consists of
1. Determine the translation vector
2. Complete the translation of CAMEL to
C’A’M’E’L’. Draw and label the image
correctly.
3. Trace the pre-image and compare it to
the image and complete the following
statements:
● CAMEL is ________ to C’A’M’E’L
4
Here is some of the language of translations.
Complete each statement
1. Point A maps to _______________.
2.
AT is translated to ____________.
3. T’ is the image of ______________.
4. △CAT is the __________________.
5. △CAT is _____________ to △C’A’T’.
6. Translations have _____________.
Translation vector:
Writing an ordered pair rule for translations
Complete the following table
The translation vector is:
Pre-Image
Vector
Image
A
B
C
Write the general ordered pair rule for the translation of a figure by vector 〈h, k〉
5
Translate each figure by the given ordered pair rule.
Write an ordered pair rule for each translation
1.
2.
(x, y) → (
,
)
If △BED is translated by 〈-1, 2〉, determine
the coordinates of △B’E’D’
Pre-Image
B
(-3, 4)
E
(1, 0)
D
(-1, 5)
(x, y) → (
,
)
If a figure has an ordered pair rule of
(x, y) → (x − 3, y + 4) , describe the
transformation.
Image
6
REFLECTIONS
Essential Question: What is the ordered pair rule for reflections over the x-axis and y-axis?
Reflections: ​A ________________________________ in which every point from a figure to its
mirror image on the other side of a line of reflection.
Note: Every point of the figure is equidistant from the line of reflection (axis of symmetry)
1. A reflection is a rigid transformation and has isometry complete the followi
● BACD is ________ to B’A’C’D
Reflect △ABC over the x-axis
Write an ordered pair rule for a reflection over
the x-axis.
7
Reflect △QRS over the y-axis
Write an ordered pair rule for a reflection over
the y-axis.
General ordered pair rule for a reflection over the x-axis:
General ordered pair rule of a reflection over the y-axis:
The following points have been reflected over the specified axis. Determine the coordinates for the
reflected points: You can use the graph if needed to determine the coordinates.
Point
Reflection
over
(-2, 3)
x - axis
(1, -1)
x - axis
(3, 4)
y - axis
(-2, -3)
y - axis
New coordinates
If a figure is 1st reflected over the x-axis then translated by 〈-1, 2〉, what would the ordered pair
rule of the sequence of transformations be:
(x, y)
Pre-Image
→
_________ →
Image 1
________
Image 2
8
Reflection over lines other that the axes
Reflection over line​ ​l
Reflection over line​ ​q
What is the equation of line​ l​ ​?
What is the equation of line​ q​ ​?
Draw the line of reflection
Draw the line of reflection
What is the equation for the line of reflection:
What is the equation for the line of reflection:
9
ROTATIONS
Essential question: ​ What are the ordered pair rules for rotations?
Rotation:
A ______________________________ that ___________________ a figure around (about)
a _____________________.
Notes: Include term angle of rotation.
Clockwise rotations
Clockwise rotations
Rotate the point ​CLOCKWISE​ the specified degrees around
the origin and complete the table.
New
Coordinates
Ordered pair rule
Rotation
(x, y )
90°
Clockwise
(-2,1)
(x, y ) →
180°
(-2,1)
(x, y ) →
270°
Clockwise
(-2,1)
(x, y ) →
Rotate the point ​COUNTERCLOCKWISE​ the specified degrees
around the origin and complete the table.
New
Coordinates
Rotation
(x, y )
Ordered pair rule
90°
Counterclockwise
(-2,1)
(x, y ) →
180°
(-2,1)
(x, y ) →
270°
Counterclockwise
(-2,1)
(x, y ) →
10
Rotate △ABC 90 clockwise around (about)
the origin
1) What is the ordered pair rule for this
rotation?
Rotate △ABC 90 counterclockwise around
(about) the origin
1) What is the ordered pair rule for this
rotation?
2) Determine the new coordinates using the
ordered pair rule.
2) Determine the new coordinates using the
ordered pair rule.
Coordinates
New Coordinates
Coordinates
A
A
B
B
C
C
3) Draw the image and label correctly
Rotate the image 180° around the origin
Ordered pair rule:
New Coordinates
3) Draw the image and label correctly
Describe each transformation:
PreImage
Image 1 Transformation
(2, 3)
(-3, 2)
(4,2)
(-4, -2)
(1,-4)
(4, 1)
(-2, -3)
(2,3)
11
COMPOSITION (SEQUENCE) OF TRANSFORMATIONS
Describe each transformation and write an ordered
pair rule:
1st Transformation
Description:
Ordered Pair Rule:
2nd Transformation
Description:
Ordered Pair Rule:
Composite ordered pair rule (Pre-image to Image 2)
12
Rotate the image clockwise 90° around the origin
then reflect Image 1 over the x-axis
Ordered pair rule for 1st Transformation:
Ordered pair rule for the 2nd Transformation:
Ordered pair rule for the composition of the two
transformations:
Translate the pre-image by ​〈-1, 2〉then rotate
Image 1 180​° around the origin
Ordered pair rule for 1st Transformation:
Ordered pair rule for the 2nd Transformation:
Ordered pair rule for the composition of the two
transformations:
Write the ordered pair rules to determine if order matters in a sequence of transformations:
Does order matter when writing rules for reflections and rotations?
(x, y )
(x, y )
13
Does order matter when writing rules for translations and reflections?
(x, y )
(x, y )
A sequence of transformations across parallel lines is the same as:
14
Graph Image 1 and Image and write the ordered
pair rules for each step of the sequence.
Graph line y = x. Then graph the
transformations.
Write an ordered pair rule for each step of the
transformation.
15
16
```
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