Name: ___________________________________
Date
12/19
Lesson
1
Mrs. White
Topic
Dilations
~Holiday Break~
1/5
2
Ratios in Similar Triangles
1/6
3
Similar Triangle Proofs
1/7
4
More Similar Triangle Proofs
1/8
5
Side Splitter etc.
1/9
6
Proving the Pythagorean Theorem
Quiz
1/12
7
Right Triangle Similarity (Geometric Mean)
1/13
8
More Geometric Mean
1/14
9
Unit 7 Review
1/15
10
Unit 7 Test
Page 1 of 24
Lesson 1: Dilations
Warm Up:
a) Map ∆𝐴𝐵𝐶 onto ∆𝑋𝑌𝑍
b) Map WXYZ onto KLMN
Dilation is a transformation that produces an image that is the same _____________ as
the pre-image but is a
, either larger or
smaller. The eyeball dilates according to the amount of available light.
A dilation changes the distance from the center of dilation, O, that maps every point P
in the plane to point P’ so that the following properties are true:
OP` = _____________
Ex 1: Given center 𝑂 and triangle 𝐴𝐵𝐶, dilate the triangle from center 𝑂 with a scale factor r = 2
Measure the following segments:
AB = ______________
A`B` = _____________
What do you notice?
**The distance between the pre-image points is also _____________________
__________________________ to get the distance between the image points**
A’B’ = k · AB, B’C’ = k · BC
and
A’C’ = k · AC
Page 2 of 24
For a dilation, we use a scale factor to change the size.
Consider 3 cases for the scale factor (k).
If ______________________ the size increases
If _______________________ the size decreases
If __________ there is no size change
Ex 2: For the given dilation to the right, determine the scale factor
AB : A’B’
4 : 12
BC : B’C’
5.5 : 16.5
AC : A’C’
5 : 15
_______
_______
________
Ex. 3: If a segment has a length of 12 and its image has a length of 4, what is the scale
factor?
Properties of a Dilation:
 A dilation will always have _________________ orientation
 The following properties are also preserved:
o Angle measure (angle measures stay the same)
o Parallelism (things that were parallel are still parallel)
o Collinearity (points on a line, remain on the line)


KEEP IN MIND, DILATION IS NOT AN ____________________ SO DISTANCE IS NOT
PRESERVED.
The only invariant point in a dilation is _____________________________________________
Negative Dilations:
Let’s think about the dilation we did before. For a dilation of 2, we
found the distance from the center of dilation and multiplied it by 2
to get the distance (along the same ray) to the image point.
Let’s do the same thing for a dilation of -1.
What do you notice about the pre-image and the image?
Page 3 of 24
A NEGATIVE dilation (scale factor = -k) will __________________ the figure _____________
around the center of dilation and also change the size.
Example 1: Negative Dilation
Example 2: Locations for the Center of Dilation
Dilations on a coordinate grid
Example: Dilate the following points according to the scale factor given. Write the coordinates of
the image.
1
a) Scale factor = 3
b) Scale factor = 2
Is there a pattern with your coordinates?
For a dilation with a scale factor of k, the coordinates of a point (x, y) will be (
,
)
Page 4 of 24
Day 2: Ratios in Similar Figures
Warm Up: If ∆QRS  ∆ZYX, identify the pairs of congruent angles and the pairs of congruent sides.
Dilations produce images that are _______________( ) to the pre-image. That is, they have the same
_________________ but not necessarily the same ______________.
Similar Polygons
DEFINITION
Two polygons are
~ polygons if and only if
1.
DIAGRAM
6
A
5
D
4
STATEMENTS
B
A 
5.4
B 
C 
C
D 
12
E
2.
F
10
H
10.8
8
AB

BC

CD

DA

G
Example 1: Identify the pairs of congruent angles and corresponding sides.
Page 5 of 24
A similarity ratio is the ratio of the __________ of the corresponding sides of two similar polygons.
(In terms of the dilation, this is the same as the reciprocal of the __________________)

The similarity ratio of ∆ABC to ∆DEF is __________ , or __________ .

The similarity ratio of ∆DEF to ∆ABC is ________ , or ____________ .
Example 3: Determine whether the polygons are similar. If so, write the similarity ratio and a
similarity statement.
Given: Rectangles ABCD and EFGH
Step 1: Identify pairs of congruent angles
Step 2: Compare corresponding sides
Step 3: Write similarity ratio and similarity statement (if applicable)
Example 4: Determine whether the polygons are similar. If
so, write the similarity ratio and a similarity statement.
Page 6 of 24
Try these on your own!
1. The accompanying diagram shows two
similar triangles. Which proportion could be
used to solve for x?
1)
2)
4. In the accompanying
diagram of equilateral
triangle ABC,
and
ABC ~ DEC . If AB is
three times as long as DE, what is the
perimeter of quadrilateral ABED?
1)
2)
3)
4)
3)
4)
20
30
35
40
2. Are the two triangles below similar? If so,
what is the similarity ratio of ∆PQR to ∆STU?
5. Tyler wants to find the height of a telephone
pole. He measured the pole’s shadow and his
own shadow and then made a diagram. What
is the height (h) of the pole?
3. In
, E is a point on
point on
,
of
and B is a
, such that ACD ~ ABE . If
, and
, find the length
.
6. Find the length of the model to the
nearest tenth of a centimeter.
Page 7 of 24
Day 3: Proving Triangles Similar
Warm Up:
1. If two triangles are similar, then corresponding sides are _____________________ and
corresponding angles are ________________________
2.
3.
In the diagram, ΔABC ~ ΔDBE. Find the value of x.
Proving Triangles Similar using Transformations:
To prove triangles are similar, you can have a sequence of dilations AND rigid motions
(___________________) that map one triangle onto the other.
Example 1: Find a sequence of transformations that will map ∆𝐷𝐸𝐹
onto ∆𝐴𝐵𝐶.
Example 2: Are the two triangles similar? Explain.
Page 8 of 24
Proving Triangles Similar without Transformations:
Just like proving triangles congruent, there are some “short cuts” to proving that two
triangles are similar so that we don’t have to prove all of the sides proportional and all of the
angles congruent.
METHOD 1: SSS SIMILARITY
Let’s focus on just the sides first.
To prove these triangles similar, we need a
series of transformations that will map one
triangle onto the other.
First, let’s take ∆𝐴𝐵𝐶 and dilate it by a scale factor of k where k is equal to
This will produce a triangle similar to ∆𝐴𝐵𝐶, called ∆𝐴`𝐵`𝐶` by definition of our
dilations.
What do we know about ∆𝐴`𝐵`𝐶`?
This is enough to prove that ∆𝐴`𝐵`𝐶` ≅ ∆𝐷𝐸𝐹 by ____________
Since ∆𝐴`𝐵`𝐶` ≅ ∆𝐷𝐸𝐹, there must be a series of rigid motions that maps one triangle
onto the other (in this case it looks like a ______________________ but in other diagrams
it could be any rigid motion!)
So, we know that ∆𝐴𝐵𝐶 ~∆𝐷𝐸𝐹 because we can map ∆𝐴𝐵𝐶 onto ∆𝐷𝐸𝐹 by first doing a
dilation and then some sequence of rigid motions.
In Summary,
If we know that the corresponding sides of two triangles
are __________________, then the two triangles are similar
Abbreviated: SSS Similarity
This theorem relies on side length. As a result, it is not used often because most of
our proofs do not give us much information about the actual lengths of our sides.
Page 9 of 24
METHOD 2: SAS SIMILARITY
The proof of this theorem is like the proof of SSS Similarity.
Now let us see if knowing two corresponding proportional
sides and the included corresponding congruent angle
(SAS) is enough for establishing similarity.
To prove ~ we need a sequence of transformations that will map one triangle onto the other.
First, let’s take ∆𝐴𝐵𝐶 and dilate it by a scale factor of k where k is equal to
This will produce a triangle similar to ∆𝐴𝐵𝐶, called ∆𝐴`𝐵`𝐶` by definition of our
dilations.
 This means that A`B` = DE and A`C` = DF because of our choice of k.
 In addition, since ∠𝐴 ≅ ∠𝐴` and the given told us ∠𝐴 ≅ ∠𝐷, we can say
____________ by _______________________
 So, ∆𝐴`𝐵`𝐶` ≅ ∆𝐷𝐸𝐹 by ____________
Since ∆𝐴`𝐵`𝐶` ≅ ∆𝐷𝐸𝐹, there must be a series of rigid motions that maps one triangle onto the
other. So, we know that ∆𝐴𝐵𝐶 ~∆𝐷𝐸𝐹 because we can map ∆𝐴𝐵𝐶 onto ∆𝐷𝐸𝐹 by first doing a
dilation and then some sequence of rigid motions.
In Summary,
If we know that two corresponding sides of two triangles are __________________
and the included angles are ___________________________
then the two triangles are similar
Abbreviated: SAS Similarity
METHOD 3: AA SIMILARITY
Once again, the proof of this theorem is like the proof of SSS ~
We will do the same as before. Use the lengths to set up a scale
factor so that we know at least one side will be congruent after
the dilation. Then, since this a dilation, we know that the angles
of the image are congruent to the pre-image and therefore congruent to the final triangle by
substitution with the given. Therefore, the triangles are congruent by __________ and
therefore ∆𝐴𝐵𝐶 ~∆𝐷𝐸𝐹 because we can map ∆𝐴𝐵𝐶 onto ∆𝐷𝐸𝐹 by first doing a dilation and then
some sequence of rigid motions.
In Summary,
If we know that two corresponding angles of two triangles are __________________
then the two triangles are similar
Abbreviated: AA Similarity
Page 10 of 24
For each example, determine the method that proves the triangles similar:
a)
b)
c)
Given: AD bisects BAE
BC  AC , BC || DE
Prove:∆𝐴𝐵𝐶~∆𝐴𝐸𝐷
B
A
D
C
E
Page 11 of 24
Day 4: More Proving Triangles Similar
Warm Up:
1. Determine the method that proves the two triangles similar.
2. Fill in the blanks below
3. Find the length of ST, if the two triangles are similar.
Proof 1:
Given: 3UT = 5RT, 3VT = 5ST
Prove: UVT  RST
Page 12 of 24
There are two other theorems that you know, that will use a lot in similarity proofs:
1. If two triangles are similar, then corresponding sides are ______________________
2. In a proportion, ______________________________________________________________
______________________________________________________________________________
(This is the formal way to say ________________________________________)
In the proportion
a c
 the values a and d are the extremes and the values b and c
b d
A
are the means.
Proof 2: In the diagram, ̅̅̅̅
𝐴𝐵 ⊥ ̅̅̅̅
𝐵𝐷, ̅̅̅̅
𝐸𝐷 ⊥ ̅̅̅̅
𝐵𝐷
B
Prove: 1. ∆𝐴𝐵𝐶 ~ ∆𝐸𝐷𝐶
1
D
C
2
2. 𝐴𝐵 ∙ 𝐷𝐶 = 𝐸𝐷 ∙ 𝐵𝐶
Statement
Reason
E
̅̅̅̅ ⊥ 𝐵𝐷
̅̅̅̅, ̅̅̅̅
̅̅̅̅
1. 𝐴𝐵
𝐸𝐷 ⊥ 𝐵𝐷
1. Given
2. ∡𝐵 𝑎𝑛𝑑 ∡𝐷 𝑎𝑟𝑒 𝑟𝑖𝑔ℎ𝑡 𝑎𝑛𝑔𝑙𝑒𝑠.
2.
3.
3.
∡𝐵 ≅ ∡𝐷
4.
4.
5.
5.
6.
6.
7.
7.
Page 13 of 24
**DETERMINE which TRIANGLES to use FIRST!**
(highlight the sides and see which triangles are formed)
Proof 3:
Page 14 of 24
Day 5: Side Splitter Theorems
Warm Up: Complete the following proofs.
Given: MN ll KL
Prove: ∆JMN ~ ∆JKL
(Leads up to Side Splitter Theorem)
Side-Splitter Theorems
Triangle Proportionality Theorem If a line
parallel to a side of a triangle intersects the
other two sides, then it divides them
proportionally.
Converse: If a line divides two sides of a
triangle proportionally, then it is parallel to the
third side.
Corollary: If three or more parallel lines
intersect two transversals, then they divide the
transversals proportionally.
These theorems give us more ways to solve similar triangle problems.
1. Find BE.
You can use similar triangles:
OR the new theorem:
Page 15 of 24
Try some on your own:
2.
a) Find US
3. In the diagram of
length of
1) 6
2) 2
3) 3
4) 15
b)Find PN
shown below,
. If
,
, and
, what is the
?
Triangle Angle Bisector Theorem (∆  Bisector Thm.)
THEOREM
HYPOTHESIS
CONCLUSION
A
An angle bisector of a ∆ divides
the opposite side into two
segments whose lengths are
B
Using the Triangle Angle Bisector Theorem
4. Find PS and SR
D
C
5. Find RV and VT
Page 16 of 24
You try it!
1) Find the length of JG.
2) Find the lengths of SR and ST
3) Suppose that an artist decided to make a larger sketch
of the trees. In the figure find LM to the nearest tenth of an
inch.
Day 6: Pythagorean Theorem proof using similar triangles
Warm Up:
Explain why the three triangles are similar.
Page 17 of 24
Since the three triangles are similar, then the
corresponding sides are proportional.
To prove the Pythagorean theorem, let’s set up some
proportions.
(It might help to draw the triangles separately)
I.
Write the proportion of the smallest triangle to the largest triangle (Using AB and AC in
the largest triangle)
II.
Write the proportion of the medium size triangle to the largest triangle (Using AB and CB
in the largest triangle)
III.
Cross multiply both proportions and show that AC2 + BC2 = AB2
Page 18 of 24
Day 7: Use geometric mean to find segment lengths in right triangles.
Warm Up: Simplify
1.
8

2.

24
3.
48
4.
80

Geometric Mean
In algebra, mean is another term that means “average.”
In geometry,
The geometric mean of two positive numbers a and b is the positive number x that satisfies
Guided Practice: Find the geometric mean of the following:
1) 2 and 8
2) 9 and 16
5) 1 and 6
4) 7 and 9
Independent Practice: Find the geometric mean of the following, round to the nearest tenth if
necessary:
1) 2 and 72
2) 4 and 25
3) 5 and 6
4) 2 and 17
What happens if you know the geometric mean?
Example: 6 is the geometric mean between 9
Example: 16 is the geometric mean between 4
and what number?
and what number?
Page 19 of 24
Mean Proportional: Altitude Rule
The altitude (perpendicular line to opposite side of the triangle) forms two
triangles are that are similar. (we’ll talked about this yesterday!)
Using similar triangles, we can get two valuable theorems:
1. Altitude Rule
2. Leg Rule (we’ll talk about this one tomorrow)
The ALTITUDE RULE
The altitude to the hypotenuse of a right triangle is the mean proportional between the
segments into which it divides the hypotenuse.
such that the altitude = CD
Example #1:
Find x.
Example #2:
Find x and y rounded to the nearest tenth.
Example #3:
The measures of the segments formed by the altitude to the hypotenuse of a right triangle are in the
ratio 1: 4. The length of the altitude is 14.
a. Find the measure of each
segment of the hypotenuse.
b. Express, in simplest radical form,
the length of each leg.
Example #4:
Find h, the height of the track, to the nearest tenth.
Page 20 of 24
Day 8: More Geometric Mean
Warm Up
Warm up: Mean Proportional: Altitude Rule
1) Solve for x.
2) Solve for x.
Mean Proportional:
3) Find the geometric mean of 2 and 72.
1) Altitude Rule (yesterday)
2) Leg Rule
Guided Practice: Find x (Review) and Find y (New)
Review: Find x, which is the altitude
New: Find y, which is a leg of the triangle
Two approaches:
a) 1st approach: Pythagorean Theorem
b) 2nd approach: Geometric Mean-Leg Rule
Page 21 of 24
Theorem:
Each leg of a right triangle is the mean proportional between the
hypotenuse and the projection of the leg on the hypotenuse.
or
Practice:
1. Find x:
2. Find x:
Leg Rule:
3. Find x:
4. Using triangle ACB below, find the following:
a) Show that AD = 4.
b) If AD = 4, then what is the length of CB?
5. Find x:
6. (a) Find CA rounded to the nearest tenth.
(b)Find CB in simplest radical form
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