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: ____________________________________________

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

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: _______________________

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 ________________________ enlargement ________________________

Example 1:

Example 2:

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 ,, BPAP 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 PEPD, 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 ABBD ⊥ and ACCD ⊥ , 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 ABBD ⊥ and ACDC ⊥ and BD = CD, then ______________________________________.

Example 2

Perpendicular Bisectors of a Triangle Activity

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 .

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 BPAP, and CP are angle bisectors of ABC∆ ,

then ______________________.

Example 3

In the diagram, N is the incenter of ABC∆ . Find EN and NB.

C

B

A

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 .

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?

____________________________________________________________________________________

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 = _________

A C

B

D

Example 2

Use the graph shown.

Find the coordinates of K, the midpoint of FH .____________

Find the length of the medianGK . ____________

Use the median GK to find the coordinates of centroid P. __________

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∆ .

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 ______________________

A C

B

D

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.

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 = _________________________________

N

Apothem PQ

P

Q

M

Central Angle MPN∠

Vocabulary

Apothem of a regular polygon

_______________________________________

Central angle of a regular polygon

___________________________________________

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