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Geometry (4102)
Isometry
Iso = same
Metry = measurement
3 Types of Isometries
Translation
Rotation
Reflection
Moving an object,
in a direction, on a plane
Translations
original
image
Same shape, different place,
same orientation
Translations
original
t
image
Properties of translations
- Same angles
- Same size
- Same length of line segments
- Same proportion of line lengths
- Same orientation
Steps to complete translations
Step 1. Measure the arrow with a ruler in
cm.
Step 2. Draw 2 or 3 dotted perpendicular
lines from the arrow.
Step 3. From each point, draw a dotted line
the same length as, and in the same
direction as the arrow.
Step 4. Label your new points and draw the
image.
Rotation
Rotating an object around
the centre of rotation
Rotating an object around the
centre of rotation
Rotations
original
image
Same shape, different place,
different orientation
Rotations
original
image
Properties of rotations
- Same shape
- Same angles
- Same length of line segments
- Same proportion of line lengths
- Different orientation
Steps to complete a rotation
Step 1. Using a ruler, measure distance from
point O to all vertices (dotted lines).
Step 2. Using a compass, measure each segment
and draw arcs.
Step 3. Using a protractor, measure angle from
point O for all vertices .
Step 4. Using a ruler, connect the vertices, and
draw the image.
Reading a protractor:
Always read from 0° to 180°
Reading a protractor
Outside numbers
• When the point of origin is
located on the right side of
the baseline, read the
numbers from left to right.
Inside numbers
• When the point of origin is
located on the left side of
the baseline, read the
numbers from right to left.
What is this angle?
What is this angle?
Align the tip of the
angle to the circle
in the middle of
the protractor
What is this angle?
Read from 0◦
upwards
What is this angle?
Read from 0◦
upwards
Angle is 62.5◦
What is this angle?
What is this angle?
90◦
What is this angle?
How do you know?
90◦
What is this angle?
90◦
What is this angle?
What is this angle?
180◦
What is this angle?
How do you know?
180◦
What is this angle?
0◦
180◦
Clockwise rotation
•
O
Counterclockwise rotation
•
O
Example:
Perform a counterclockwise rotation r of
∆ABC through 120o about point O
A
O •
B
C
Step 1. Using ruler, measure distance from
point O to all vertices
A
O •
B
C
Step 1. Using ruler, measure distance from
point O to all vertices
A
O •
B
C
Step 2. Using compass, measure each
segment and draw arcs counterclockwise
A
O •
B
C
Step 2. Using compass, measure each
segment and draw arcs counterclockwise
A
O •
B
C
Step 2. Using compass, measure each
segment and draw arcs counterclockwise
A
O •
B
C
Step 3. Using protractor, measure 120o
(R to L) from point O for all vertices
B
A
O •
B
C
Step 3. Using protractor, measure 120o
(R to L) from point O for all vertices
C
A
O •
B
C
Step 3. Using protractor, measure 120o
(R to L) from point O for all vertices
A
A
O •
B
C
Step 4. Using ruler, connect the vertices,
and label them as A', B', C'
C'
C
A'
B'
O •
A
B
C
Reflection
Reflection on the line of
reflection
Reflection on the line of reflection
Reflections
original
image
Same shape, different place,
different orientation
Reflections
original
image
Properties of reflections
- Same shape
- Same angles
- Same length of line segments
- Same proportion of line lengths
- Different orientation
Steps to complete a reflection
Step 1. Draw perpendicular dotted lines from
vertices to line of reflection.
Step 2. Using a ruler, measure each segment.
Step 3. Using the same distances, draw new
points on the other side of the line.
Step 4. Using a ruler, connect the vertices, and
draw the image.
Example:
Reflect triangle ABC in line s
A
C
B
s
Step 1. Draw dotted lines from
vertices to line of reflection
Make sure they are
perpendicular to
the line!
A
C
B
s
Step 1. Draw dotted lines from
vertices to line of reflection
A
C
B
s
Step 1. Draw dotted lines from
vertices to line of reflection
A
C
B
s
Step 2. Using ruler,
measure each segment
A
2.5 cm
1.5 cm
C
B
2 cm
s
Step 3. Using the same distances, draw
new points on the other side of the line
A
2.5 cm
A'
1.5 cm
C
B
2 cm
s
A
Step 3. Using the same distances, draw
new points on the other side of the line
A
2.5 cm
A'
1.5 cm
C
B
B'
2 cm
s
B
Step 3. Using the same distances, draw
new points on the other side of the line
A
2.5 cm
A'
2 cm
C'
C
B
B'
1.5 cm
s
C
Step 4. Using ruler, connect the vertices,
and label them as A', B', C'
A
A'
C'
C
B
B'
s
Summary of Isometry
Translation
t
Rotation
•O
Reflection
s
Size transformations
h
Proportional shape, same orientation
Size transformations
h
Pg. 2.2
Enlarge quadrilateral ABCD
under size transformation h
(ratio k=2)
Enlarge quadrilateral ABCD under
size transformation h (ratio k=2)
A
B
D
C
Enlarge quadrilateral ABCD under
size transformation h (ratio k=2)
A
Make
bigger!
B
D
C
Step 1. Pick a point O somewhere
near the shape
A
B
•
D
C
This point can be anywhere
near the shape. There are
many possible places.
O
Step 1. Pick a point O somewhere
near the shape
A
B
•
O
D
C
This point can be anywhere
near the shape. There are
many possible places.
Centre of
similitude
Step 2. Draw lines from point O to
each of the vertices, and beyond
A
B
•
D
C
O
Step 2. Draw lines from point O to
each of the vertices, and beyond
A
B
•
D
C
O
Step 2. Draw lines from point O to
each of the vertices, and beyond
A
B
•
D
C
O
Step 3. Measure the distance from
point O to each of the vertices
4 cm
A
B
•
D
C
O
Step 3. Measure the distance from
point O to each of the vertices
4 cm
A
•
5 cm
B
D
C
O
Step 3. Measure the distance from
point O to each of the vertices
4 cm
A
•
5 cm
B
D
C
4.5 cm
O
Step 3. Measure the distance from
point O to each of the vertices
4 cm
A
•
5 cm
B
D
C
2 cm
4.5 cm
mOA = 4 cm
mOB = 5 cm
mOC = 4.5 cm
mOD = 2 cm
O
Step 4. Multiply the measurement
by the ratio k
4 cm
A
B
•
O
D
C
k=2
A
4 cm x 2 = 8 cm
Step 5. Starting from point O, measure the new
distance on the line, and mark the point as A'
8 cm
A
• A'
B
•
O
D
C
k=2
A
4 cm x 2 = 8 cm
Step 4. Multiply the measurement
by the ratio k
A
• A'
•
O
5 cm
B
D
C
k=2
B
5 cm x 2 = 10 cm
Step 5. Starting from point O, measure the new
distance on the line, and mark the point as B'
A
• A'
B
• B'
10 cm
•
O
D
C
k=2
B
5 cm x 2 = 10 cm
Step 4. Multiply the measurement
by the ratio k
A
• A'
•
B
• B'
O
D
C
4.5 cm
k=2
C
4.5 cm x 2 = 9 cm
Step 5. Starting from point O, measure the new
distance on the line, and mark the point as C'
A
• A'
B
O
D
C
• B'
•
C
•
9 cm
k=2
C'
4.5 cm x 2 = 9 cm
Step 4. Multiply the measurement
by the ratio k
A
• A'
B
D
O
2 cm
C
• B'
•
D
•
k=2
C'
2 cm x 2 = 4 cm
Step 5. Starting from point O, measure the new
distance on the line, and mark the point as D'
A
• A'
•
B
D
C
• B'
•
•
D
O
D'
4 cm
k=2
C'
2 cm x 2 = 4 cm
Step 6. Connect the points to form
quadrilateral A'B'C'D'
A
• A'
B
•
D
C
• B'
•
•
D'
C'
Ta-da!
O
So when k = 2, the object
becomes twice as large...
Pg. 2.5
Reduce quadrilateral ABCD
under size transformation h
(ratio k=1/2)
Reduce quadrilateral ABCD under
size transformation h (ratio k=1/2)
A
B
D
C
Step 1. Pick a point O somewhere
near the shape
A
•
O
B
D
C
This point can be anywhere
near the shape. There are
many possible places.
Step 2. Draw lines from point O to
each of the vertices, and beyond
A
•
B
D
C
O
Step 2. Draw lines from point O to
each of the vertices, and beyond
A
•
B
D
C
O
Step 2. Draw lines from point O to
each of the vertices, and beyond
A
•
B
D
C
O
Step 3. Measure the distance from
point O to each of the vertices
8 cm
A
•
B
D
C
O
Step 3. Measure the distance from
point O to each of the vertices
8 cm
A
•
10 cm
B
D
C
O
Step 3. Measure the distance from
point O to each of the vertices
8 cm
A
•
10 cm
B
9 cm
C
D
O
Step 3. Measure the distance from
point O to each of the vertices
8 cm
A
•
O
10 cm
B
9 cm
C
D
4 cm
mOA = 8 cm
mOB = 10 cm
mOC = 9 cm
mOD = 4 cm
Step 4. Multiply the measurement
by the ratio k
8 cm
A
•
O
B
D
k = 1/2
C
A
8 cm x 1/2 = 4 cm
Step 5. Starting from point O, measure the new
distance on the line, and mark the point as A'
4 cm
A
• A'
•
O
B
D
k = 1/2
C
A
8 cm x 1/2 = 4 cm
Step 4. Multiply the measurement
by the ratio k
A
• A'
•
O
10 cm
B
D
k = 1/2
C
B
10 cm x 1/2 = 5 cm
Step 5. Starting from point O, measure the new
distance on the line, and mark the point as B'
A
• A'
•
O
5 cm
•
B'
B
D
k = 1/2
C
B
10 cm x 1/2 = 5 cm
Step 4. Multiply the measurement
by the ratio k
A
• A'
•
•
O
B'
9 cm
B
D
k = 1/2
C
C
9 cm x 1/2 = 4.5 cm
Step 5. Starting from point O, measure the new
distance on the line, and mark the point as C'
A
• A'
•
•
O
B'
•
C'
4.5 cm
B
D
k = 1/2
C
C
9 cm x 1/2 = 4.5 cm
Step 4. Multiply the measurement
by the ratio k
A
• A'
•
•
O
B'
•
C'
4 cm
B
D
k = 1/2
C
D
4 cm x 1/2 = 2 cm
Step 5. Starting from point O, measure the new
distance on the line, and mark the point as D'
A
• A'
•
B'
•
C'
•
•
D'
O
2 cm
B
D
k = 1/2
C
D
4 cm x 1/2 = 2 cm
Step 6. Connect the points to form
quadrilateral A'B'C'D'
A
• A'
•
B'
•
C'
•
•
D'
B
D
C
Ta-da!
O
So when k = 1/2,
the object becomes
half as large...
Rules for transformations
If k < 1, the image is smaller than the
original
If k > 1, the image is larger than the original
What happens if k = 1?
What happens if k = 1?
The image is the same size
as the original
Steps to complete a transformation
Step 1. Pick a point O somewhere near the shape
Step 2. Draw lines from point O to each of the
vertices, and beyond
Step 3. Measure the distance from point O to each
of the vertices
Step 4. Multiply the measurement by the ratio k
Step 5. Starting from point O, measure the new
distance on the line, and mark the point as '
Step 6. Connect the points to form the image
Properties of transformations
Same angles
Same proportion of line lengths
Same order of points
Parallel and perpendicular lines kept
Pg. 2.7
Draw the image of triangle
ABC under the size
transformation h with centre
O and ratio k = –2.
Draw the image of triangle ABC under the
size transformation h with centre O and
ratio k = –2.
A
B
C
Step 1. Pick a point O somewhere
near the shape
A
•
B
O
C
Step 2. Draw lines from each of the
vertices to point O, and beyond
A
•
B
O
C
Step 2. Draw lines from each of the
vertices to point O, and beyond
A
•
B
O
C
Step 2. Draw lines from each of the
vertices to point O, and beyond
A
•
B
O
C
Step 3. Measure the distance from
point O to each of the vertices
1.7 cm A
•
B
O
C
Step 3. Measure the distance from
point O to each of the vertices
1.7 cm A
•
3.2 cm
B
O
C
Step 3. Measure the distance from
point O to each of the vertices
1.7 cm A
•
3.2 cm
B
O
1.2 cm C
mOA = 1.7 cm
mOB = 3.2 cm
mOC = 1.2 cm
Step 4. Multiply the measurement
by the ratio k
1.7 cm A
•
B
O
C
A
k = –2
1.7 cm x –2 = –3.4 cm
Step 4. Multiply the measurement
by the ratio k
When k is
negative, the
image is on the
other side of
point O
1.7 cm A
•
B
O
C
A
k = –2
mOA'= 1.7 cm x –2 = –3.4 cm
Step 5. Starting from point O, measure the new
distance on the other side of the line,
and mark the point as A'
A
•
B
O
• A'
A
C
3.4 cm
k = –2
mOA'= 1.7 cm x –2 = –3.4 cm
Step 4. Multiply the measurement
by the ratio k
A
•
3.2 cm
B
O
• A'
B
C
k = –2
mOB'= 3.2 cm x –2 = –6.4 cm
Step 5. Starting from point O, measure the new
distance on the other side of the line,
and mark the point as B'
A
6.4 cm
•
•
B'
O
• A'
B
B
C
k = –2
mOB'= 3.2 cm x –2 = –6.4 cm
Step 4. Multiply the measurement
by the ratio k
A
•
B
O
• A'
C
1.2 cm C
k = –2
mOC'= 1.2 cm x –2 = –2.4 cm
Step 5. Starting from point O, measure the new
distance on the other side of the line,
and mark the point as C'
•
C' 2.4 cm A
•
•
B'
O
• A'
C
B
C
k = –2
mOC'= 1.2 cm x –2 = –2.4 cm
Step 6. Connect the points to form
the image
•
C'
A
•
•
B'
O
• A'
B
C
Yay!
Rules for transformations
If 0 < k < 1, the image is smaller than the
original
If k > 1, the image is larger than the original
If k < 0, the image is on the other side of
point O
Similar vs. Congruent
triangles
proportional
exactly the same
Similar vs. Congruent
triangles
proportional
exactly the same
Similar vs. Congruent
triangles
proportional
exactly the same
Similar vs. Congruent
triangles
Two triangles are congruent
when there is an isometry
between them
Two triangles are congruent
when there is an isometry
between them
Isometry = same measurement
(Translation, Rotation, Reflection)
Pg. 3.4
Draw a triangle with one angle
o
35 with the two sides
forming this angle 5cm and
7cm respectively
Step 1. Draw a line of 7cm
(draw longer side first)
7 cm
Step 2. At one end of the line, using
the protractor, draw an angle of 35o
7 cm
Step 2. At one end of the line, using
the protractor, draw an angle of 35o
Remember to align
the end of the line
with the centre of
the protractor
7 cm
Step 3. Using the ruler, draw a line
through the angle of 35o
35o
7 cm
Step 4. Using the ruler, measure 5cm
along this new side
5 cm
35o
7 cm
Step 5. Using the ruler, connect the two lines
5 cm
35o
7 cm
Step 6. Outline the triangle
5 cm
35o
7 cm
Done!
Draw a triangle with one angle 35o with the two
sides forming this angle 5cm and 7cm
respectively
5 cm
35o
7 cm
What if you had drawn the triangle with the
sides switched?
5 cm
35o
7 cm
Same triangle, but rotated
That’s okay!
7 cm
5 cm
Draw a triangle with one angle 35o with the two
sides forming this angle 5cm and 7cm
respectively
5 cm
35o
7 cm
When given two side lengths and an angle,
there is only one way to draw the triangle
S-A-S
side side
angle
5 cm
35o
7 cm
When given two side lengths and an angle,
there is only one way to draw the triangle
First property of congruent triangles
S-A-S
5 cm
side side
angle
35o
7 cm
When given two side lengths and an angle,
there is only one way to draw the triangle
S-A-S Property
Two triangles are congruent if two
sides and the contained angle of one
triangle are congruent to the
corresponding parts of the other
Steps to draw a S-A-S triangle
Step 1. Draw the longer side first
Step 2. At one end of the line, using the
protractor, draw the angle
Step 3. Using the ruler, draw a line through the
angle
Step 4. Using the ruler, measure the second
length along this new side
Step 5. Using the ruler, connect the two lines
Step 6. Outline the triangle
Pg. 3.1
Draw any sort of triangle with
a 5cm side contained between
o
o
an angle of 30 and one of 40
Step 1. Draw a line of 5cm
5 cm
Step 2. Using the protractor, draw the
angle of 30o
Remember to align
the end of the line
with the centre of
the protractor
5 cm
Step 2. Using the protractor, draw angle of
30o from one end
Remember to align
the end of the line
with the centre of
the protractor
5 cm
Step 3. Using the ruler, draw a line through
the angle of 30o
30o
5 cm
Step 4. On other end of line, use the
protractor to draw an angle of 40o
Remember to align
the end of the line
with the centre of
the protractor
30o
5 cm
Step 5. Using the ruler, draw a line
through the angle of 40o
30o
40o
5 cm
Step 6. Connect the points to form the
triangle
30o
40o
5 cm
Done!
Draw any sort of triangle with a 5cm
side contained between an angle of
30o and one of 40o
30o
40o
5 cm
What if you had drawn the triangle
with the angles on the other sides?
30o
40o
5 cm
Same triangle, but reflected
That’s okay!
30o
40o
5 cm
Draw any sort of triangle with a 5cm side
contained between an angle of 30o and
one of 40o
30o
40o
5 cm
When given a side length and two angles,
there is only one way to draw the triangle
A-S-A
side
angle angle
30o
40o
5 cm
When given a side length and two angles,
there is only one way to draw the triangle
Second property of congruent triangles
A-S-A
side
angle angle
30o
40o
5 cm
When given a side length and two angles,
there is only one way to draw the triangle
A-S-A Property
Two triangles are congruent if two
angles and the included side of one
triangle are congruent to the
corresponding parts of the other
Steps to draw an A-S-A triangle
Step 1. Draw the side
Step 2. Using the protractor, draw the first angle
from one end
Step 3. Using the ruler, draw a line through the
first angle
Step 4. Using the protractor, draw the second
angle from the other end
Step 5. Using the ruler, draw a line through the
second angle
Step 6. Connect the points to form the triangle
Pg. 3.6
Draw any sort of triangle with
a 6cm side, a 5 cm side and a
3 cm side
Step 1. Draw a line of 6cm
(draw the longest side first)
6 cm
Step 2. Using ruler and compass,
measure out 5 cm from one end
and draw an arc
5 cm
6 cm
Step 2. Using ruler and compass,
measure out 5 cm from one end
and draw an arc
5 cm
6 cm
Step 3. Using ruler and compass,
measure out 3 cm from the other end
and draw an arc
3 cm
6 cm
Step 3. Using ruler and compass,
measure out 3 cm from the other end
and draw an arc
3 cm
6 cm
Step 4. Make a point where the two
arcs intersect
6 cm
Step 5. Draw lines connecting the
three vertices
6 cm
Step 5. Draw lines connecting the
three vertices
6 cm
Step 6. Outline your triangle
5 cm
3 cm
6 cm
What if you had drawn the triangle
with the sides switched?
5 cm
3 cm
6 cm
Same triangle, but rotated
That’s okay!
6 cm
5 cm
3 cm
Draw any sort of triangle with a 6cm
side, a 5 cm side and a 3 cm side
5 cm
3 cm
6 cm
When given three side lengths, there is
only one way to draw the triangle
S-S-S
side side side
5 cm
3 cm
6 cm
When given three side lengths, there is
only one way to draw the triangle
Third property of congruent triangles
S-S-S
side side side
3 cm
5 cm
6 cm
When given three side lengths, there is
only one way to draw the triangle
S-S-S Property
Two triangles are congruent if the
three sides of one triangle are
congruent to the corresponding sides
of the other
Steps to draw a S-S-S triangle
Step 1. Draw a line (draw the longest side first)
Step 2. Using ruler and compass, measure out
the second line from one end and draw an arc
Step 3. Using ruler and compass, measure out
the third line from one end and draw an arc
Step 4. Make a point where the two arcs
intersect
Step 5. Draw lines connecting the three vertices
Step 6. Outline your triangle
Two triangles are similar
when there is a similarity
between them
Two triangles are similar
when there is a similarity
between them
Similarity = proportional sides
and same angles
Let’s look at these triangles a bit
more...
•
C'
A
•
•
B'
O
• A'
B
C
Comparing these two triangles
A'
A
B'
B
C
C'
Comparing the angles
A'
A
B'
B
C
C'
A   A'
B  B'
 C  C'


Comparing the angles
A'
A
B'
B
C
A   A'
B  B'
 C  C'


C'
This means these triangles have the
same corresponding angles
Comparing the sides
A'
mA'B'
A
B'
mAB
B
C
C'
mA'B' = mB'C' = mA'C'
mAB mBC mAC
Comparing the sides
A'
mA'B'
A
B'
mAB
B
C
C'
mA'B' = mB'C' = mA'C'
mAB mBC mAC
This means these
triangles have the
same proportional
sides (their ratios
are constant)
Comparing the sides
A'
mA'B'
A
B'
mAB
B
C
C'
mA'B' = mB'C' = mA'C'
mAB mBC mAC
Ratio of similitude
Are these triangles similar or
congruent?
A'
A
B'
B
C
C'
Are these triangles similar or
congruent?
A'
A
B'
B
C
C'
Are these triangles similar or
congruent?
A'
A
B'
B
C
C'
Why?
Why are these triangles similar?
A'
A
B'
B
C
C'
1) They have the same angles
2) The corresponding sides are proportional
First property of similar triangles
A'
A
B'
B
C
C'
If the measures of two angles of two triangles are
known and the corresponding angles are
congruent, the triangles are similar
A-A Property
angle angle
A'
A
B'
B
C
C'
Two triangles are similar if they have
two congruent corresponding angles
Second property of similar triangles
A'
5 cm
A
2.5
B' cm
5 cm
C'
6 cm
2.5 cm
B
C
3 cm
If the lengths of three sides of two triangles are
known, and the lengths of the corresponding sides
are proportional, the triangles are similar
S-S-S Property
side side side
A'
5 cm
A
2.5
B' cm
5 cm
C'
6 cm
2.5 cm
B
C
3 cm
Two triangles are similar if the
three corresponding sides are proportional
Third property of similar triangles
A'
5 cm
A
B
B'
C'
6 cm
2.5 cm
C
3 cm
If two pairs of corresponding sides are proportional
and the contained angles are congruent, the
triangles are similar
S-A-S Property
side side
angle
A'
5 cm
A
B
B'
C'
6 cm
2.5 cm
C
3 cm
Two triangles are similar if they have a congruent
angle contained between two corresponding
proportional sides
Remember:
proportional
exactly the same
Similar vs. Congruent
triangles
Pg. 4.2. Two similar triangles and the
lengths of their sides
7.5 cm
4.5 cm
2.5 cm
1.5 cm
2 cm
6 cm
original
image
How did the original become the image?
7.5 cm
4.5 cm
2.5 cm
1.5 cm
2 cm
6 cm
original
image
How did the original become the image?
Transformation
7.5 cm
4.5 cm
2.5 cm
1.5 cm
2 cm
6 cm
original
image
Two similar triangles (transformation)
k=?
7.5 cm
4.5 cm
2.5 cm
1.5 cm
2 cm
6 cm
original
image
length of side (image) = 7.5 cm = 3
length of side (original) 2.5 cm
k=3
7.5 cm
4.5 cm
2.5 cm
1.5 cm
2 cm
6 cm
original
image
length of side (image) = 6 cm = 3
length of side (original) 2 cm
k=3
7.5 cm
4.5 cm
2.5 cm
1.5 cm
2 cm
6 cm
original
image
length of side (image) = 4.5 cm = 3
length of side (original) 1.5 cm
k=3
7.5 cm
4.5 cm
2.5 cm
1.5 cm
2 cm
6 cm
original
image
Find the missing dimensions of the image
?
2.5 cm
?
1.5 cm
2 cm
original
4 cm
image
Step 1. Find k (ratio of similitude)
?
2.5 cm
?
1.5 cm
2 cm
original
4 cm
image
Step 1. Find k (ratio of similitude)
length of side (image) = 4 cm = 2 = k
length of side (original) 2 cm
?
2.5 cm
?
1.5 cm
2 cm
original
4 cm
image
Step 2. Multiply each original side by k to find
each length in the image
length of side (image) = ? cm = 2
length of side (original) 1.5 cm cross-multiply
? cm = 2 (1.5 cm) = 3 cm
?
2.5 cm
? = 3 cm
1.5 cm
2 cm
original
4 cm
image
Step 2. Multiply each original side by k to find
each length in the image
length of side (image) = ? cm = 2
length of side (original) 2.5 cm cross-multiply
? cm = 2 (2.5 cm) = 5 cm
? = 5 cm
2.5 cm
? = 3 cm
1.5 cm
2 cm
original
4 cm
image
Pg. 4.3 Similar triangles ABC and A'B'C'
A
A'
B
C
B'
C'
Pg. 4.3 Similar triangles ABC and A'B'C'
Congruent corresponding angles
Proportional sides
A
A'
B
C
B'
C'
Similar triangles ABC and A'B'C'
Since angles A and A' are congruent,
sides BC and B'C' must be proportional
A
A'
B
C
B'
C'
Similar triangles ABC and A'B'C'
Since angles B and B' are congruent,
sides AC and A'C' must be proportional
A
A'
B
C
B'
C'
Similar triangles ABC and A'B'C'
Since angles C and C' are congruent,
sides AB and A'B' must be proportional
A
A'
B
C
B'
C'
Similar triangles ABC and A'B'C'
k = mA'B' = mA'C' = mB'C'
mAB
mAC
mBC
A
A'
B
C
B'
C'
Pg. 4.5 Similar triangles ADE and ABC
A
D
B
E
C
Similar triangles ADE and ABC
A
D
B
E
C
Similar triangles ADE and ABC
A
D
B
E
C
Similar triangles ADE and ABC
Since angle A is in both
triangles, sides DE and BC
are corresponding sides
A
D
B
E
C
Similar triangles ADE and ABC
Since angle A is in both
triangles, sides DE and BC
are corresponding sides
Since sides DE and BC
are corresponding sides,
they are parallel
A
D
B
E
C
Similar triangles ADE and ABC
Since angle A is in both
triangles, sides DE and BC
are corresponding sides
A
D
B
Since sides DE and BC
are corresponding sides,
they are parallel
Since sides DE and BC
are parallel, angles
D and B, and E and C
are congruent
E
C
Similarity and congruency can
be used on other shapes too!
Pg. 5.3 Given the similar
quadrilaterals below. The ratio of
similitude between
quadrilaterals ABCD and A'B'C'D'
is 3/2.
Four-sided
shapes
Same
corresponding
angles
Proportional
corresponding
sides
Pg. 5.3 Given the similar
quadrilaterals below. The ratio of
similitude between
quadrilaterals ABCD and A'B'C'D'
is 3/2.
=k
ABCD = original
A'B'C'D' = image
Given the similar quadrilaterals below. The ratio of
similitude between quadrilaterals ABCD and A'B'C'D'
is 3/2.
A'
A
1.9 cm
3.7 cm
C'
B
D
2.3 cm
C
D'
3.3 cm
k = 3/2
B'
What is the length of each of the sides of
quadrilateral A'B'C'D'?
A'
A
1.9 cm
3.7 cm
B'
B
D
2.3 cm
C
D'
3.3 cm
k = 3/2
C'
What is the length of each of the sides of
quadrilateral A'B'C'D'?
mAB = 1.9 cm
mBC = 2.3 cm
mCD = 3.3 cm
mAD = 3.7 cm
x (3/2)
= 2.85 cm = mA'B'
= 3.45 cm = mB'C'
= 4.95 cm = mC'D'
= 5.55 cm = mA'D'
A'
A
1.9 cm
3.7 cm
B'
B
D
2.3 cm
C
D'
3.3 cm
k = 3/2
C'
What is the length of each of the sides of
quadrilateral A'B'C'D'?
mAB = 1.9 cm
mBC = 2.3 cm
mCD = 3.3 cm
mAD = 3.7 cm
x (3/2)
= 2.85 cm = mA'B'
= 3.45 cm = mB'C'
= 4.95 cm = mC'D'
= 5.55 cm = mA'D'
A'
A
1.9 cm
2.85 cm
5.55 cm
3.7 cm
B'
B
D
2.3 cm
C
3.3 cm
D'
3.45 cm
4.95 cm
k = 3/2
C'
What is the perimeter of quadrilateral A'B'C'D'?
A'
A
1.9 cm
2.85 cm
5.55 cm
3.7 cm
B'
B
D
2.3 cm
C
3.3 cm
D'
3.45 cm
4.95 cm
k = 3/2
C'
What is the perimeter of quadrilateral A'B'C'D'?
Perimeter = Sum of all the lengths of the sides
A'
A
1.9 cm
2.85 cm
5.55 cm
3.7 cm
B'
B
D
2.3 cm
C
3.3 cm
D'
3.45 cm
4.95 cm
k = 3/2
C'
What is the perimeter of quadrilateral A'B'C'D'?
Perimeter A'B'C'D' = 2.85cm + 3.45cm + 5.55cm + 4.95cm
= 16.8 cm
A'
A
1.9 cm
2.85 cm
5.55 cm
3.7 cm
B'
B
D
2.3 cm
C
3.3 cm
D'
3.45 cm
4.95 cm
k = 3/2
C'
Is the ratio of the perimeters of the quadrilaterals
equal to the ratio of similitude?
Perimeter A'B'C'D' = 2.85cm + 3.45cm + 5.55cm + 4.95cm
= 16.8 cm
A'
A
1.9 cm
2.85 cm
5.55 cm
3.7 cm
B'
B
D
2.3 cm
C
3.3 cm
D'
3.45 cm
4.95 cm
k = 3/2
C'
Is the ratio of the perimeters of the quadrilaterals
equal to the ratio of similitude?
Perimeter A'B'C'D' = 2.85cm + 3.45cm + 5.55cm + 4.95cm
= 16.8 cm
Perimeter ABCD = 1.9 cm + 2.3 cm + 3.3 cm + 3.7 cm
= 11.2 cm
A'
A
2.85 cm
5.55 cm
1.9 cm
3.7 cm
B'
B
D
2.3 cm
C
3.3 cm
D'
3.45 cm
4.95 cm
k = 3/2
C'
Is the ratio of the perimeters of the quadrilaterals
equal to the ratio of similitude?
Perimeter A'B'C'D' = 16.8 cm = 3 = k
Perimeter ABCD = 11.2 cm 2
Yes!
A'
A
1.9 cm
2.85 cm
5.55 cm
3.7 cm
B'
B
D
2.3 cm
C
3.3 cm
D'
3.45 cm
4.95 cm
k = 3/2
C'
Pg. 5.5
The sides of two regular pentagons
have a ratio of similitude equal to
10/3. The perimeter of the large
pentagon is 150 cm. What is the
length of a side of the small
pentagon?
k = 10/3
Five equal sides
Pg. 5.5
The sides of two regular pentagons
have a ratio of similitude equal to
10/3. The perimeter of the large
pentagon is 150 cm. What is the
length of a side of the small
pentagon?
The sides of two regular pentagons have a ratio of similitude
equal to 10/3. The perimeter of the large pentagon is 150 cm.
What is the length of a side of the small pentagon?
The sides of two regular pentagons have a ratio of similitude
equal to 10/3. The perimeter of the large pentagon is 150 cm.
What is the length of a side of the small pentagon?
k = 10/3
original
image
The sides of two regular pentagons have a ratio of similitude
equal to 10/3. The perimeter of the large pentagon is 150 cm.
What is the length of a side of the small pentagon?
Perimeter (image) = 150 cm
Regular pentagon = 5 equal sides
Each side of pentagon (image)
= 150 cm ÷ 5 = 30 cm
k = 10/3
original
image
The sides of two regular pentagons have a ratio of similitude
equal to 10/3. The perimeter of the large pentagon is 150 cm.
What is the length of a side of the small pentagon?
Perimeter (image) = 150 cm
Regular pentagon = 5 equal sides
Each side of pentagon (image)
= 150 cm ÷ 5 = 30 cm
30 cm
k = 10/3
original
image
The sides of two regular pentagons have a ratio of similitude
equal to 10/3. The perimeter of the large pentagon is 150 cm.
What is the length of a side of the small pentagon?
Side length (image) = 30 cm = 10 = k
Side length (original) = x cm
3
30 cm
k = 10/3
original
image
The sides of two regular pentagons have a ratio of similitude
equal to 10/3. The perimeter of the large pentagon is 150 cm.
What is the length of a side of the small pentagon?
Side length (image) = 30 cm = 10 = k cross-multiply
Side length (original) = x cm
3
30 cm
k = 10/3
original
image
The sides of two regular pentagons have a ratio of similitude
equal to 10/3. The perimeter of the large pentagon is 150 cm.
What is the length of a side of the small pentagon?
30 cm = 10
x cm 3
3 (30 cm) = 10 (x cm)
90 cm = 10x
10
10
30 cm
9 cm = x
k = 10/3
original
image
Scale diagrams
= Ratio of similitude
=k
Size in picture
Actual length
Scale diagrams
Size in picture  Actual length
Size in picture : Actual length
Size in picture Actual length
Scale diagrams
Bedroom
no. 3
12 cm
Bedroom Bathroom
no. 2
C
C
Bedroom
no. 1
Hall
C
Dining
room
Hall
Living
room
Pg. 6.3 Size of picture on paper
C
C
8 cm
Kitchen
Bedroom
no. 3
12 m
Bedroom Bathroom
no. 2
C
C
Bedroom
no. 1
Hall
C
Dining
room
Hall
Living
room
Pg. 6.3 Actual plan of a house
C
Kitchen
C
8 m
1 unit
on the picture
1,000,000 units
in real life
Scale 1 1,000,000
For any units
(Same units for both)
Size in picture
Actual length
Scale diagrams
Size in picture  Actual length
Size in picture : Actual length
Size in picture Actual length
House diagram
Length of house in picture = 12 cm
Actual length of house = 12 m
Always goes from picture to actual object
House diagram
Length of house in picture = 12 cm
Actual length of house = 12 m
Scale: 12 cm to 12 m Remember: 1 m = 100 cm
12 cm = 12 cm = 1
12 m
1200 cm
100
House diagram
Length of house in picture = 12 cm
Actual length of house = 12 m
Scale: 12 cm to 12 m Remember: 1 m = 100 cm
12 cm = 12 cm = 1
The scale of the
12 m
1200 cm 100
house picture is
1 to 100
Living room
Length of living room in picture = 7 cm
Actual length of living room = ?
Scale of house: 1 to 100
1 = 7 cm
100
?
Living room
Length of living room in picture = 7 cm
Actual length of living room = ?
Scale of house: 1 to 1000
1 = 7 cm Picture length
100
?
Actual length
Living room
Length of living room in picture = 7 cm
Actual length of living room = ?
Scale of house: 1 to 1000
1 = 7 cm cross-multiply
100
?
100 (7 cm) = 1 (?)
700 cm = (?)
Living room
Length of living room in picture = 7 cm
Actual length of living room = ?
Scale of house: 1 to 1000
1 = 7 cm cross-multiply
100
?
The living room is
100 (7 cm) = 1 (?)
actually 700 cm, or
7 m, long
7000 cm = (?)
Steps to solve scale problems
To find the scale:
Step 1. Measure the length(s) in the picture
Step 2. Compare one picture length to its
corresponding real-life length (make sure
units are the same!)
Step 3. Reduce the fraction to its lowest form
(use calculator)
Steps to solve scale problems
To find the length in real life:
Step 1. Find the scale, if it is not given
Step 2. Measure the length in the picture
Step 3. Compare picture to real life (make
sure units are the same!) and crossmultiply
Steps to find scale and similarity
Step 1. Measure the drawing that is given
Step 2. Find the corresponding height of the
other picture
Step 3. Using scale, find actual height of object
Size in picture
Actual length
Scale diagrams
If Size in picture ˃ Actual length, enlargement
If Size in picture = Actual length, same size
If Size in picture ˂ Actual length, reduction