Ch. 11 - Similarity
Class Notes
What Is Similarity?
Warm-Up
• Each pair, open textbook to p. 564
• Each person use patty paper, protractor
and ruler to complete steps 1 & 2.
Mark corners, A, B etc.
• Mark all measures of sides and angles
on patty paper.
• Glue paper into your notebook. In ten
minutes, you will be quizzed on this.
Quiz Question 1
• Whiteboard, marker & rag in table.
• Hold up answer, facing forward, 1 min.
Q: Are these shapes congruent?
A. Yes, they have the same angles.
B. Yes, they can be superimposed.
C. No, they have the same angles but not
the same sides.
D. Cannot be determined. (CBD)
Notes
Polygons are similar if,
• Their corresponding angles are
congruent (same) and,
• Their corresponding sides are
proportional. That is, you can
multiply each side by the same
scale factor to get the
corresponding side length.
Warm-Up, Part II (5 minutes)
• Calculate the ratio of each pair of
corresponding sides in these
shapes. Record on patty paper.
• Example: RS/CD = 4.1cm/2.3cm
ratio RS/CD = 1.8
• In five minutes, you will be quizzed.
Quiz Question 2
• Whiteboard, marker & rag in table.
• Hold up answer, facing forward, 1 min.
Q: Are these shapes similar?
A. Yes, they can be superimposed.
B. No, side lengths not all the same.
C. Yes, they have the same angles and all
pairs of corresponding sides are
proportional.
D. Cannot be determined. (CBD)
Quiz Question 3
• Whiteboard, marker & rag in table.
• Hold up answer, facing forward, 5 min.
–Draw three similar, but not congruent
triangles.
–Mark all side lengths.
–Mark which angles & sides are
congruent.
–Draw one non-similar triangle with side
lengths.
Notes
• Dilation – Process of growing or
shrinking. Example: Eye pupils dilate in dark.
• Scale factor – How much the preimage has grown/shrunk to create the
image. Example: 3:1 = scale factor 3
• Calculate scale factor by dividing
length of corresponding sides:
image/pre-image.
• Calculate scale factor by dividing
length of corresponding sides
image/pre-image.
• Pre-image 3 cm -> Image 9 cm
= 9/3 = scale factor 3
• Pre-image 8 cm -> Image 2 cm
= 2/8 = scale factor ¼ = 0.25
• Scale factor > 1 is growing ; < 1 is
shrinking
Polygons are similar if,
• Their corresponding angles are
congruent (same) and,
• Their corresponding sides are
proportional, have the same scale
factor one shape to the next.
• The ratio of sides within each
shape is the same: small-tomedium-to-big.
Quiz Question 1 (1 minute)
Q: Are these shapes,
A. Congruent
B. Similar
C. Congruent & Similar
D. Neither
• Are corresponding angles congruent?
YES. (90o)
• Are corresponding sides proportional?
NO.
• Conclusion – These polygons are not
similar or congruent.
Quiz Question 2
• Answer # 1, p. 568 on whiteboard.
Quiz Question 3
• Answer # 8, p. 568 on whiteboard.
Quiz Question 4
• Answer # 5, p. 568 on whiteboard.
• Also write the scale factor.
Cleanup Grade
• Start at 100% each quarter.
• Keep credit if:
– Push in chair to touch table.
– Pick up trash near desk, even if not yours.
– Remove trash inside desk.
– Erase whiteboard.
– Return textbook, whiteboard, marker, rag into
desk.
• Also start with 100% for following rules…
• HW p. 568 # 1-10
• # 1 & 2 Find two shapes that have
same angles and corresponding
parts, but have different sizes.
• # 3 – 5 Draw original shape on graph
paper. Then repeat but make all
sides bigger by same amount. (2x?)
Drawing Dilations
1.
2.
3.
4.
5.
Warm-Up
Place a point on the left of a new page in your
notebook.
Draw three rays ‘radiating’ out from this
point, to the upper, middle & lower right.
Randomly place an additional point on each
ray. Connect them to form a triangle.
Place a second point on each ray, at twice the
distance as the first. Connect to make a
second triangle.
What do we notice about the two triangles?
Quiz Question 1 (1 minute)
Q: What is the relationship
between these triangles?
A. Congruent
B. Similar
C. Congruent & Similar
D. Neither
Quiz Question 2 (1 minute)
Q: What is the scale factor?
A. 0.5
B. 1.0
C. 2.0
D. None of the above
Quiz Question 3 (1 minute)
Q: Corresponding sides in
these two triangles are:
A. congruent
B. parallel
C. perpendicular
D. None of the above
Dilations Centered on Same Point
Dilations Centered on Same Point
Dilations work for all polygons…
Center of dilation can be
inside shapes.
Scale factor can be negative…
What is the scale factor in
this dilation?
Similar Shapes Centered on Same
Point of Dilation...
• Corresponding angles are congruent.
• Corresponding sides are parallel .
• Every pair of corresponding sides has the same
ratio to each other.
• Ratios of center-to-vertex distances are the
same for corresponding vertices in each
shape.
• Ratios between sides within the same shape,
are identical for pre-image and image.
How to Know if
Shapes are Similar
& Prove It?
• Have HW open in first minute of
class to earn credit.
• Open textbook to p. 568.
Warm-Up
• You will be quizzed on the following
graph in 10 minutes.
• On your personal whiteboard graph:
(3 ,2) (4, -2) (-2, -3)
• Dilate triangle with scale factor of 2,
Draw rays centered on origin.
• Write a coordinate rule for this.
Similar Shapes
• Corresponding angles are congruent.
• Corresponding sides are parallel
(if share the same center of dilation.)
• Pairs of corresponding sides have the
same ratio to each other, old & new…
AB to A’B’; BC to B’C’…
Scale Factor = new length/old lengths
Transformation rules for dilations
only when centered on the origin (0,0)
• 2x dilation: (x, y) → (2x, 2y)
• 1/3x dilation: (x, y) → (1/3x, 1/3y)
• Stretch - shapes not similar!
(x, y) → (2x, y) Growing wider only.
• Another stretch:
(x, y) → (2x, 3y) Growing differently
in x & y directions.
Similar Shapes
• Ratio between sides within the same
shape, small:medium:large, are
identical for pre-image and image.
• Ratio of center of dilation-to-vertex
distances are the same for
corresponding vertices in each
shape.
If all corresponding angles remain
congruent, must sides be proportional?
•
•
•
Are corresponding angles congruent? YES. (90o)
Are corresponding sides proportional? NO.
Conclusion – These polygons are not similar or
congruent.
If all sides are proportional, must
corresponding angles remain congruent?
• Scale factor consistently 18/12=1.5
• But…are corresponding angles
congruent? NO!
Are these Shapes Similar?
Are these Shapes Similar?
• Corresponding angles congruent?
• Corresponding sides proportional?
• Conclusion?
Are these Shapes Similar?
• Corresponding angles congruent? YES.
• Corresponding sides proportional? YES.
• Conclusion – These polygons are
similar.
CORN ~ PEAS
For Triangles only…
Angle-Angle Triangle Similarity Shortcut
• If any two angles in two triangles are
congruent…then both triangles must be
similar! (AA Shortcut)
• Do not need to measure third pair of angles
(Triangle Sum Conjecture says they must be
the same.)
• Do not need to measure sides or calculate
scale factor… sides must all grow/shrink by
same ratio to keep angles congruent.
HW
• p. 574 # 1-10
How to Indirectly Measure
Heights of Tall Objects
with Similar Triangles
Indirectly Measure Heights of Tall
Objects with Similar Triangles
• You will measure the height of the…
1. Gold ball on flagpole outside
2. Ceiling by vending machines
3. Bottom edge of U.S. flag in
commons
4. Water tower (tricky; can’t measure
directly to base…use similar
triangles, twice?)
Method #1
The Mirror Method
Indirectly Measure Heights of Tall
Objects with Similar Triangles
•
•
•
•
String, knots 1 meter apart
30 cm ruler
Mirror with taped edges
When done, leave all supplies neatly
on desk
• You assigned mirror is numbered
Measuring Height w Similar Triangles
Measuring Height w Similar Triangles
Angle-Angle Triangle Similarity Shortcut
AA(A) proves similarity – screaming helps!
Measuring Height w Similar Triangles
Angle-Angle Triangle Similarity Shortcut
AA(A) proves similarity – screaming helps!
Scale factor =
5m/2m = 2.5
5m
2m
Measuring Height w Similar Triangles
Angle-Angle Triangle Similarity Shortcut
AA(A) proves similarity – screaming helps!
3.75m
Scale factor =
5m/2m = 2.5
Scale factor x small height =
2.5 x 1.5m = 3.75m big height
1.5 m
5m
2m
Method #2a
The Held-Out Ruler Method
Measuring Height or Distance
with Similar Triangles
Method #2b
The Better Held-Out Ruler Method
Method #3
The Shadow Method
Measuring Height w Similar Triangles
Measuring Height w Similar Triangles
3-day project
Indirectly Measure Heights of Tall
Objects on Campus
with Similar Triangles
Indirectly Measure Heights of Tall
Objects with Similar Triangles
• Each partner records data & calculations.
• Measure each height two ways - mirror &
ruler methods:
1. Gold ball on flagpole outside
2. Ceiling in commons
3. Bottom edge of U.S. flag in commons
4. Water tower (tricky; can’t measure
directly to base…use similar triangles,
twice?)
Friday 1/9 – Measure Heights of Tall
Objects with Similar Triangles
Mirror
Ruler
You are graded on clean up &
returning materials intact…
Place mirror, meter stick, little ruler
on desk to be checked in.
Place mirror, meter stick, little ruler
on desk to be checked in.
HW
• HW due tomorrow 11.3 #1-10.
• Quiz tomorrow, sections 1-3.
• May use your notes.
• May NOT use cell phone
calculator for scale factors.
Indirect measurement, using shadows
4.0 cm flag pole shadow
on Google Maps
1) Calculate scale factor
from flag & tower
shadows
2) Scale factor x known
flag height = tower
height
14.0 cm tower shadow
on Google Maps, from
base
Unit Review
Partners Warm-Up – 10 min
• On whiteboard, graph polygon:
A (3, 4), B (2, -4), C (-5, -4)
• Draw dilations centered on the origin
with scale factors of 2:1 final:initial
(2x bigger) and 1:2 (1/2x smaller).
Move vertices along rays…
• Label A’, A’’…
Solo – 10 min
• On graph paper in your notebook, graph
polygon:
A (-3, -4), B (-2, 4), C (5, -4)
• With center of dilation on point D (3, 4),
• Dilate 2:1 final:initial (2x)
• 1:2 (1/2x)
• -2:1 (-2x) Hint: Draw -1 dilation
first
Solo – 10 min
• On graph paper in your notebook,
graph polygon:
• Complete steps for question 2,
p. 617.
Solo – 10 min
• On graph paper in your notebook,
graph polygon:
• Complete steps for question 1,
p. 617.
• Discuss with group, write down your
conclusion and be ready to explain
your conclusions.
Partners Warm-Up – 10 min
• On graph paper in your notebook,
graph polygon:
A (-3, -4), B (-2, 4), C (5, -4)
• Draw 2:1 (2x) and 1:2 (1/2x) and -2:1
(-2x) dilations centered on point O
(3, 4). Move vertices along rays…
Agenda
• QUIZ – Yes: notes & calculator. No: cell phones.
Review for Test
1. In notebook, graph (3, 2) (4, -1) (-3, 3) (-2, -5).
Tape in graph paper if needed.
2. Draw rays from origin through each vertex.
3. Dilate above shape with scale factors of 0.5
and 2.
4. Write as complete sentences:
“In a dilation, _________ angles ___________.”
“We can control the scale factor by
_____________________________________.”
Review for Test
1. In notebook, graph (3, 2) (4, -1) (-3, 3) (-2, -5).
Tape in graph paper if needed.
2. Draw rays from origin through each vertex.
3. Dilate above shape with scale factors of 0.5
and 2.
4. Write as complete sentences:
“In a dilation, corresponding angles remain congruent.”
“We can control the scale factor by increasing the
distance between center-of-dilation & vertices by that factor.”
Dilation using ‘ray method’
1. Draw a circle with your compass.
2. Place a point outside the circle for
the center of dilation.
3. Use the ‘ray method’ to dilate the
circle 2x bigger. Don’t need a graph.
4. Dilate the original circle ¾ of original
size, using only compass & straight
edge.
Dilations Centered on Same Point
Dilations work for all polygons…
Center of dilation can be
inside shapes.
Center of dilation can be
outside shapes.
Dilations Centered on Same Point
Scale factor can be negative…
What is the scale factor in
this dilation?
1. Compare polygon (3, 2) (4, -1) (-3, 3) (-2, -5)
and polygon (5, 0) (6, -3) (-1, 1) (0, -7).
2. Write as complete sentence:
“These polygons are: congruent/similar/neither
(may be more than one) because __________
_____________________.
3. “If similar, the scale factor is ____________.”
1. Compare polygon (3, 2) (4, -1) (-3, 3) (-2, -5)
and polygon (1.5, 2) (2, -1) (-1.5, 3) (-1, -5).
2. Write as complete sentence:
“These polygons are: congruent/similar/neither
(chose one) because _____________________.
3. “If similar, the scale factor is ____________.”
1. Compare polygon (3, 2) (4, -1) (-3, 3) (-2, -5)
and polygon (1.5,1) (2,-0.5) (-1.5,1.5) (-1,-2.5).
2. Write as complete sentence:
“These polygons are: congruent/similar/neither
(chose one) because _____________________.
3. “If similar, the scale factor is ____________.”
Friday 1/16
TEST – Similarity & Dilations
•
•
•
•
30 min
No notes
No cell phone
Yes calculator
•
•
•
•
•
10/10 = A+
9/10 = A7/10 = B
5/10 = C
3/10 = D
A level topics 11.4 & 11.5
• Verify geometric properties of dilations
WarmUp – 5 min
• Draw a square 2 cm on each side.
• Draw a square 6 cm on each side.
• Draw dashed lines to show how many little
squares fit in the big square.
• Calculate the area of each and write in each
square with units.
• Fill in this Conjecture:
“If corresponding sides of two similar
polygons compare in a ratio of m/n, then
their areas compare in the ratio of
______________.”
Proportional Volume Conjecture
• Draw a cube 2 cm on each side.
• Draw a bigger cube 6 cm on each side.
• Draw dashed lines to show one little cube in the
corner of the big cube.
• Calculate the volume of each cube.
• Fill in this Conjecture:
“If corresponding edges (or radii or heights)
of two similar solids compare in a ratio of
m/n, then their volumes compare in the
ratio of ______________.”
For shapes with a scale factor of 2,
how do these ‘scale up’?
• Perimeter?
• Area?
For solids with scale factor of 2,
how do these ‘scale up’?
• Surface Area?
• Volume?
For shapes with a scale factor of 3,
how do these ‘scale up’?
• Perimeter?
• Area?
For solids with scale factor of 3,
how do these ‘scale up’?
• Surface Area?
• Volume?
Worksheet to Complete
Lesson 11.5 – Proportions with Area and Volume
1) Which two HW Q’s would you most
like to see?
2) Working with your partner, read
pp. 599-602
‘Why Do Elephants Have Big Ears?’
Discuss, then write your answers for
Q’s 1-15. Show Mr. Sidman.
1.
2.
3.
4.
5.
6.
7.
8.
You will need your compass.
Place a point in the center of the notebook page.
Draw three rays ‘radiating’ out from this point.
Randomly place one additional point on each ray.
Place them at the edges of the paper. Connect
them to form a BIG triangle.
Place a second point at half the distance along each
ray as the first. This makes a similar triangle with a
scale factor of 0.5 compared to the first.
Bisect one corresponding side of each triangle.
Construct one corresponding median for each
triangle. Find the ratio of big-to-little medians.
Bisect a corresponding angle in each. Find the ratio
of the big-to-little angle bisector segments.
Conclusion
• Corresponding (matching) dimensions of
similar triangles all have the same scale factor
(ratio):
little side = little median = little angle bisector
big side
big median
angle bisector
little altitude = little midsegment =
big altitude
big midsegment
little perpendicular bisector
big perpendicular bisector
1.
2.
3.
4.
5.
6.
7.
8.
You will need your compass.
Place a point in the center of the notebook page.
Draw three rays ‘radiating’ out from this point.
Randomly place one additional point on each ray.
Place them at the edges of the paper. Connect
them to form a BIG triangle.
Place a second point at half the distance along each
ray as the first. This makes a similar triangle with a
scale factor of 0.5 compared to the first.
Find the ratio of the small-to-big perimeters.
For one corresponding angle in each triangle, drop a
perpendicular bisector to the opposite side.
Use this altitude (height) to find the ratio of areas.
WarmUp – 5 min
1. Write a step-by-step proof that
∆LMN ̴ ∆EMO.
2. What is length y?
1. Draw two rays forming an acute angle.
2. On one ray, use a ruler to mark off lengths 8 cm
and then an additional 10 cm from vertex. Label
these segments.
3. On the other ray, mark off segments 12 cm and
an additional 15cm.
4. Connect points to make a little triangle inside
the big.
5. Are these triangles similar?
6. Calculate the ratio 8cm to 10cm. 12cm to 15cm.
What do you notice?
7. What else do you notice about these triangles?