Unit 9: Transformations, Triangles, and Area
Lesson 9.1 Perform Congruence Transformations
Lessons 4.8, 9.3, and 9.4 from textbook
Objectives
•
Represent and model transformations such as translation and reflection in the coordinate plane and
describe the results.
Transformation _______________________________________________________________________
Translations
You can describe a translation by the notation
(x, y) → _____________________________
a = ______________________
b = ______________________
Example 1
Use coordinate notation to describe the translation.
5 units up, 6 units down ___________________.
Example 2
Figure ABCD has the vertices A(-4, 3), B(-2, 4) C(-1, 1),
and D(-3, 1). Sketch ABCD and its image after the
transformation (x, y) → (x + 5, y – 2).
Example 3
Complete the statement using the description of the translation. In the description, points (2, 0)
and (3, 4) are two vertices of a triangle.
If (2, 0) translates to (-3, 3), then (3, 4) translates to ___________________.
REFLECTIONS
Definition: _____________________________________________
Example 4
Coordinate Rules for Reflections
If (a, b) is reflected in the x-axis, its image is the point _____________________.
If (a, b) is reflected in the y-axis, its image is the point _____________________.
If (a, b) is reflected in the line y = x, its image is the point _____________________.
If (a, b) is reflected in the line y = -x, its image is the point _____________________
Reflection Theorem
__________________________________________
Example 5
Reflect the segment in the line
a) y = x
b) y = -x
ROTATIONS
Definition: ____________________________________________
Coordinate Rules for Rotations
When a point (a, b) is rotated counterclockwise about
the origin, the following are true:
1. For a rotation of 90o: ________________________
2. For a rotation of 180o: _______________________
3. For a rotation of 270o: _______________________
Example 6
Rotate the parallelogram 270o about the origin. List the vertices
of the pre-image and image.
Pre-image vertices: ______________________________
Image vertices: _________________________________
Rotation Theorem
__________________________________________
Example 7
Unit 9: Transformations, Triangles, and Area
Lesson 9.2 Perform Similarity Transformations
Lessons 6.7 from textbook
Objectives
•
Show and describe the results of dilations in the coordinate plane.
Dilation _________________________________
Center of dilation _________________________________
Scale factor of dilation _________________________________
___________________________________, where k is the scale factor.
Reduction ________________________
Example 1:
Example 2:
enlargement ________________________
Example 3:
Example 4:
Example 5:
Unit 9: Transformations, Triangles, and Area
Lesson 9.3: Compositions of Transformations and Symmetry
Lessons 9.5 – 9.6 from textbook
Objectives:
•
Perform a glide reflection in the coordinate plane using a combination of a translation and a
reflection.
• Identify and construct lines of symmetry for two-dimensional objects.
• Determine whether an object has line symmetry or rotational symmetry.
GLIDE REFLECTIONS
Reflection: ____________________________
Transformation: ____________________________
COMPOSITION OF TRANSFORMATIONS
Composition Thereom
The composition of two (or more) isometries is an isometry. (Ex: glide reflection).
Example 1
The endpoints of RS are R(1, -3) and S(3, 4). Graph the
Image of RS after the composition.
Reflection: in the y-axis
Rotation: 90o about the origin
LINE SYMMETRY
Definition: ________________________________________________
Line of symmetry ____________________________________________________________________
Example 2
How many lines of symmetry does each triangle have?
______________
________________
_____________
ROTATIONAL SYMMETRY
Definition: ___________________________________________
_____________________________________________________
Example 3
Determine whether the figure has rotational symmetry. If it does, describe the rotations that map the
figure on to itself.
____________________
Example 4
Construct a quadrilateral with two lines of symmetry.
_______________________
Unit 9: Transformations, Triangles, and Area
Lesson 9.4: Use Perpendicular Bisectors
Lesson 5.2 from textbook
Objectives
•
• Use properties of perpendicular and angle bisectors to identify equal distances.
Use properties of perpendicular bisectors to locate the point of concurrency of a triangle.
Vocabulary
perpendicular bisector _________________________________________________________________
concurrent __________________________________________________________________________
point of concurrency __________________________________________________________________
circumcenter ________________________________________________________________________
CP is a ⊥ bisector of AB.
AP ≅ __________
Perpendicular Bisector Theorem
In a plane, if a point is on the perpendicular bisector of a segment,
then it is ______________________________________________.
If CP is a ⊥ bisector of AB , then __________________________.
Converse of the Perpendicular Bisector Theorem
In a plane, if a point is equidistant from the endpoints of the segment,
Then it is ________________________________________________.
If DA = DB, then D lies on the ⊥ bisector of AB.
Example 1
BD is the perpendicular bisector of AC . Find AD.
Example 2
In the diagram, WX is the perpendicular bisector of YZ.
A) What segment lengths in the diagram are equal? ________________
B) Is V on WX ? Explain. ____________________________________
Perpendicular Bisectors of a Triangle Activity
1) Using the given scrap paper, fold it in half from top to bottom into two equal sections.
2) Draw a scalene triangle in each of the sections. Make the triangles different from each other.
3) Cut out each triangle using the scissors.
4) For each triangle, fold one end of the triangle over to one of the
other ends. Mark this crease with your pen. This creates a perpendicular
bisector of one of the sides of the triangle.
5) Repeat this process two more times for each triangle, creating a perpendicular bisector for all three
sides of the triangle.
4) Do the three bisectors intersect at one point? ____________________________________
5) Label the vertices of the triangle A, B, and C. Label the point of intersection of the perpendicular
bisectors as P. Measure AP, BP, and CP.
What do you observe? _________________________________________
Concurrency of Perpendicular Bisectors Theorem
The perpendicular bisectors of a triangle intersect at a
point that is equidistant form the vertices of the triangle.
If PD, PE and PF are perpendicular bisectors, then ______________________.
Example 3
In the diagram, the perpendicular bisectors of ∆ABC meet at point G
and are the solid lines. Find the indicated measures.
DA = __________
AB = ____________
BG = ____________
GC = __________
BE = ____________
EC = ____________
Unit 9: Transformations, Triangles, and Area
Lesson 9.5: Use Angle Bisectors
Lesson 5.3 from textbook
Objectives
•
• Use properties of angle bisectors to identify distances relationships.
Use properties of angle bisectors to locate the point of concurrency of a triangle.
Vocabulary
Angle bisector _______________________________________________________________________
PS is an angle bisector of ∠QPR.
∠QPS ≅ __________
Angle Bisector Theorem
In a plane, if a point is on the bisector of an angle,
then it is ______________________________________________.
If AD is an angle bisector of ∠BAC and BD ⊥ AB and CD ⊥ AC , then ________________________.
Example 1
Find the measure of <GFH.
Find the value of x.
Converse of the Angle Bisector Theorem
If a point is in the interior of an angle and is equidistant from the sides,
of the angle, then it is ___________________________________________.
If BD ⊥ AB and DC ⊥ AC and BD = CD, then ______________________________________.
Example 2
Perpendicular Bisectors of a Triangle Activity
A
1) Using a protractor and a ruler, draw the angle bisectors
of ∆ABC .
2) What do you notice about the angle bisectors?
______________________________________________
3) Label the point of intersection of the angle bisectors as P.
This point is known as the incenter of the triangle.
Draw a line segment that represents the distance from P to
side AB , and label this segment PD .
C
4) Repeat this process: Draw a line segment that represents
the distance from P to side BC and side AC and label these
segments PE and PF respectively.
5) What can you conclude about PD, PE, and PF? _________________________________________
Concurrency of Angle Bisectors Theorem
The angle bisectors of a triangle intersect at a
point that is equidistant form the vertices of the triangle.
If AP, BP and CP are angle bisectors of ∆ABC ,
then ______________________.
Example 3
In the diagram, N is the incenter of ∆ABC . Find EN and NB.
B
Unit 9: Transformations, Triangles, and Area
Lesson 9.6: Use Medians and Altitudes
Lesson 5.5 from textbook
Objectives
• Use properties of medians to find measures of segments in triangles formed by the intersection of
medians.
• Use properties of altitudes to find the measures of segments in a triangle formed by the
intersection of altitudes.
• Use a protractor and straightedge to locate the medians and altitudes of a triangle.
Vocabulary
Median of a triangle___________________________________________________________________
Centriod ____________________________________________________________________________
Alititude of a triangle __________________________________________________________________
Orthocenter _________________________________________________________________________
BD is a median of ∆ABC .
1. Draw the other two medians of ∆ABC . Label them CF and AE .
B
2. The three medians are ___________________________.
3. The point of concurrency of ∆ABC is called the ___________________.
Label this point P.
4. Find the following measures:
AP = _________
PE = __________
5. What do you notice about the measure AP and PE?
A
D
____________________________________________________________________________________
Concurrency of Medians of a Triangle
The medians of a triangle intersect at a point that is _________________
the distance from each vertex to the midpoint of the opposite side.
AP = __________
BP = ____________ CP = _____________
Example 1
In ∆RST , Q is the centroid, SQ = 8, RW = 10, and QV = 3.
Find the following measures:
SQ = ________
QW = ________
WT = _________
RT = ________
RQ = ________
RV = _________
C
Example 2
Use the graph shown.
Find the coordinates of K, the midpoint of FH .____________
Find the length of the median GK . ____________
Use the median GK to find the coordinates of centroid P. __________
B
BD is an altitude of ∆ABC .
1. Draw the other two altitudes of ∆ABC . Label them CF and AE .
2. The three altitudes are ___________________________.
3. The point of concurrency of ∆ABC is called the ___________________.
Label this point P.
4. P is located on the __________________ of ∆ABC .
A
D
Concurrency of Altitudes of a Triangle Theorem
The lines containing the altitudes of a triangle are concurrent.
Example 3
Find the orthocenter P in a right and acute triangle.
Classification: ______________________
Classification: __________________________
Location of orthocenter ____________________
Location of orthocenter ______________________
C
Unit 9: Transformations, Triangles, and Area
Lesson 9.7: Areas of Regular Polygons
Lesson 11.6 from textbook
Objective
•
• Find the area of a triangle using its derived formula.
Find the area of regular polygons by finding the number of triangles that create each polygon
and the formula for area of a triangle.
M
Vocabulary
Q
Apothem PQ
N
Apothem of a regular polygon
_______________________________________
P
Central angle of a regular polygon
Central Angle ∠MPN
___________________________________________
Central Angle Theorem
The measure of the central angle of a regular polygon is _____________________________________
What kind of triangles form regular polygons?
1. Draw the diagonals of the octagon.
2. How many triangles are there? ____________________
3. Find the measure of one central angle. __________________
4. What kind of congruent triangles form a regular octagon? _____________________
Area of a Regular Polygon Theorem
A = ______________________________________
Example 1
You are decorating the top of a table by covering
it with small ceramic tiles. Find the area of top
of the table.
A = _________________________________
Example 2
Find the area of each regular polygon.
A = _____________________________
A = ________________________
Example 3
A regular nonagon is inscribed in a circle with a radius
of 4 units. Find the perimeter and area of the nonagon.
P = _____________________
A = _____________________
Example 4
Find the area and perimeter of each the regular polygon.
A = ______________
P = ______________
Example 5