Grade 10 Essential Math Unit 6: Similarity of Figures
Learning Goal(s)/Topics:
1) demonstrate an understanding of transformations on a 2-d shape, including:
a. translations
b. rotations
c. Reflections
d. Dilations (similarity)
** Translations, rotations, and reflections are not covered in the textbook. Do the worksheets indicated
to review these concepts.**
2) demonstrate an understanding of primary trigonometric ratios (sine, cosine, tangent) by:
a. applying similarity to right triangles
b. generalizing patterns from similar right triangles
c. solving problems
Learning Activities:
Complete the following questions, and make sure to check your answers at the back of the textbook.
There is an answer key there. Make sure to ask questions in class if you are not getting the right
answers.
____ Read “Transformation Review Sheet”
____ Translation, Rotation, Reflection worksheet 1
http://www.superteacherworksheets.com/geometry/translation-rotation-reflection-1_TZQTQ.pdf
____ Translation, Rotation, Reflection worksheet 2
http://www.superteacherworksheets.com/geometry/translation-rotation-reflection-2_TZQTM.pdf
____Section 6.1 Similar Polygons (work book)
____ Section6 .2 Determining if Two Polygons are Similar (work book)
____ Section 6.3 Drawing Similar Polygons (work book)
____Section 6.4 Similar Triangles (work book)
____Review (work book)
Assessment:
This unit will have two assessment pieces – a project and a test. The test will be based on the review.
You will hand in the project by the date of the test. The test and the project will each be worth 5% of
your final grade.
______________ Unit Test date
_____________ _ Unit Project due
Transformation Review Sheet
Transformations
Plane figures and solids can be changed (or transformed) by translation, reflection and dilation. The
symmetry of shapes is related to translation, reflection and rotation.
Reflection
Triangle ABC and its reflection A'B'C' have the same size and shape, that is they are congruent.
Translation
A translation is defined by specifying the distance and the direction of a movement. For
example, triangle ABC is translated by 2 units to the right.
Triangle ABC and its image A'B'C' have the same size and shape. That is, they are congruent.
Rotation
Rotation is defined by stating the centre of rotation, amount of turning in degrees and the
direction of rotation (clockwise or anticlockwise). For example, triangle ABC is rotated about O
through 90º in an anticlockwise direction.
Triangle ABC and its image A'B'C' have the same size and shape, that is they are congruent.
We notice that:
The reflection, translation and rotation are congruent transformations.
Dilation
If a figure is enlarged or reduced and retains its shape, then it is said to be dilated. This is an
aspect of similarity as shown below.
Note that the stretching (or shrinking) of a shape is called a dilation. It is clear that dilation is
not a congruent transformation, because the size of the shape is changed.
In general:


Lengths and areas are not preserved under dilation.
If the dilation factor is the same for each side of the figure, then the figures are similar.
(Source: http://www.mathsteacher.com.au/year8/ch10_geomcons/12_trans/trans.htm )
Unit 5 Project: Finding the Height of a Tree (not the textbook project)
Objective:
Your assignment is to find the height of tree on campus. You will calculate the height
using two different methods, and then compare the answers that you get using the two
methods.
Method
Method 1) Shadow Reckoning:
Go to your tree and measure the shadow cast by the tree. Then measure the length of
the shadow of a student from your group, who stands by the tree (but not in it's
shadow). Measure the student's height. Then draw a sketch of the tree and the student,
label the sketch with the measurements, and calculate the height of the assigned
object.
Draw the tree, the student, and both shadows on a piece of poster board. Explain
every step of your work, and write these explanations clearly and completely,
including all of equations and any theorems that you use. This explanation will be
part of your "poster".
Method 2) Mirror Method:
Use a small hand mirror (a locker mirror will do). Place the mirror on the ground
about 10 or 15 feet from the base of the tree. Then, looking in the mirror, walk
forwards or backwards until you can see the top of the tree centered in the mirror,
Measure the distance from where you are standing to the center of the mirror, and the
distance from the center of the mirror to the base of the tree. Use similar triangles to
calculate the height of the tree.
\
In each of the two methods, explain every step of this project. Your explanation must
contain labelled diagrams, measurements, and a complete explanation of the steps in
this project, including any and all theorems used.
Comparison:
Compare the answers you got. If you found somewhat different heights from you the
two methods, explain why this might have happened. Explain why the triangles are
similar, in each case. How might these methods be useful in other situations? Do you
think this might be how people in earlier times might have measured objects too tall to
measure directly?
(Based on:
http://mathforum.org/~sanders/exploringandwritinggeometry/similartriangles.htm )
Gr.10 Essential Math Unit 5 Project Rubric: Finding the Height of a Tree
Criteria
Method 1 sketch
0
Not submitted
1
Sketch includes
few of the
requirements
2
Sketch includes
some of the
requirements
3
Sketch
requirements:

Tree,
student
and
shadows
drawn
 labelled
measureme
nts for all
Method 1
calculation
Not submitted
Calculation has
many errors
and/or hard to
follow steps
Calculation has
few errors and
easy to follow
steps
Calculation has:



Method 2 sketch
Not submitted
Sketch includes
few of the
requirements
Sketch includes
some of the
requirements
easy to
follow steps
correct
answer
based on
measuremen
ts
Clear
explanation/
connection
to math
theorems
from this
unit (similar
triangle
connections)
Sketch
requirements:

Tree,
student ,
mirror
 labelled
measureme
nts for all
Method 2
calculation
Not submitted
Calculation has
many errors
and/or hard to
follow steps
Calculation has
few errors and
easy to follow
steps
Calculation has:



Comparison
easy to
follow steps
correct
answer
based on
measuremen
ts
Clear
explanation/
connection
to math
theorems
from this
unit (similar
triangle
connections)
Answer all of the following questions: (4 possible marks)
1) If you found somewhat different heights from you the two
methods, explain why this might have happened.
2) Explain why the triangles are similar, in each case.
3) How might these methods be useful in other situations?
4) Do you think this might be how people in earlier times might
have measured objects too tall to measure directly?
_____ 16 = ______ %