Transcript Algebra I - Denise Kapler

Unit 3
Similar Triangles
Agenda
Similar Triangles
similar triangles have corresponding angles
I know That
that are congruent and that corresponding sides
are proportional. Similar polygons have the
same shape but may have different sizes based
on the similarity ratio. Order Matters in naming,
correspondence and in determining the ratio.
When creating the similarity ratio, match up the
English words. See the following.
?
How is similar different than congruent?
Symbol for Similar is ~
Word Similarity Ratio
November 2014
ΔABC ~ ΔXYZ
Use English words to determine the ratio:
AB is to XY as BC is to YZ
AB is to XY as CA is to ZX
𝑆𝑚𝑎𝑙𝑙
𝐿𝑎𝑟𝑔𝑒
=
1
2
𝐿𝑎𝑟𝑔𝑒
𝑆𝑚𝑎𝑙𝑙
=
2
1
Agenda
Similar Triangles
the rules for similar triangles apply to all polygons:
I know 1. Corresponding angles are congruent
?
2. Corresponding side lengths are proportional
3. Order matters
What is an analogy?
Analogy presents a logic based relationship between two
pairs of words or numbers. You can write it 3 ways.
Examples: Open is to shut as small is to ______
Copper : Metal :: Oxygen : Non-metal
5
6
5:6::30:36 , =
Word
𝒂 𝒄
=
𝒃 𝒅
30
36
And
a:b::c:d
a is to b as c is to d
11 7 7
: :: :
2 4 8 16
7
1
2= 8
7
1
4 16
November 2014
Agenda
Similar Triangles
there are 3 ways to solve a proportional
I know That
relationship:
1. Up and down – do a quick check for factors
2. Across – do a quick check for factors
3. Cross Products – always works
When working with a proportional relationship,
order matters – it must be preserved. See next.
?
When working with a proportional
relationship why does order matter?
Word
𝒂
𝒃
=
𝒄
= scale factor = similarity ratio
𝒅
November 2014
Agenda
Similar Triangles
I know That similar triangles exist when at least 2
?
angles are congruent.
Given two triangles with two angles that are
congruent, are the 3rd angles congruent as well?
How do you know beyond a shadow of a doubt?
Write a similarity statement for
these two triangles.
Word AA – 2 congruent angles
November 2014
Agenda
Similar Triangles
similar triangles have 3 corresponding side
I know That
lengths that are proportional.
?
What the similarity ratio of the triangle pair #1?
Of pair #2?
Word SSS – 3 proportional sides
November 2014
Agenda
I know
?
Similar Triangles
That two triangles are similar if 2 side lengths
are proportional and one set of corresponding
angles are congruent. .
Write a similarity statement for the given
triangles. What is the similarity ratio of the
side lengths?
Solve for a and x.
SAS – 2 proportional sides
Word 1 congruent angle pair
November 2014
Agenda
Similar Triangles
The similarity ratio is determined by creating an analogy in
English words and converting to a proportional math
relationship.
Small polygon’s shorter side
is to the Larger polygon’s
shorter side as the Small
polygon’s longest side is to the
Larger polygon’s longest side.
1:2::5:10
𝟏
𝟐
=
𝟓
𝟏𝟎
Large triangle’s shorter side is
to the Smaller polygon’s
shorter side as the Large
polygon’s longest side is to the
Smaller polygon’s longest side.
4:2::7:3.5
𝟒
𝟐
=
𝟕
𝟑.𝟓
=
𝟐
𝟏
November 2014
Similar Triangles
Agenda
the properties of similar triangles and
I know That
proportional relationships facilitate the solving of
many problems. These properties include:
a. Triangle Proportionality Theorem
b. Two-transversal Proportionality
c. Triangle Bi-sector Theorem
?
Word Facilitate the Solving
December 2014
Similar Triangles
Agenda
if the similarity ratio of two similar figures is
I know That
, then the ratio of their perimeters is , and the
𝑎
𝑏
ratio of their areas is
𝑎 2
𝑏
or
𝑎2
.
𝑏2
𝑎
𝑏
the similarity ratio of two similar
? Iffigures
is 4:6::small:large. Given
Asmall = 12cm2, determine the Alarge.
Word
For a closed 2D figure:
Perimeter – distance around.
Area – square units inside
December 2014
These two triangles are similar with sides in the ratio 2:1
(the sides of one are twice as long as the other):
What can we say about their areas?
The answer is simple if we just draw in three more lines:
We can see that the small triangle fits into the big
triangle four times. So when the lengths are twice as
long, the area is four times as big
So the ratio of their areas is 4:1
We can also write 4:1 as 22:1
Similar Triangles
Agenda
I know The best way to study for a math exam is
by working problems: examples from
book, and those reviewed in class and
completed for homework.
?
What have I mastered?
What do I need to work on still?
Word Dedicated to the learning.
December 2014
Math Review November 24, 2014
1. Given
MNPQ ~RSTU
Find the measure
of ∠z. Find the
measure of x.
Similar ~
1. Corresponding
Sides are
proportional.
2. Corresponding
Angles ≅
2. Simplify the
radical. 48
To simplify a radical
means to find
another expression
with the same
value. It
does not mean to
find a decimal
approximation.
3. Graph
y=(x+1)(x+1)
Identify the roots.
Write in standard
form.
Standard
Quadratic
y= Ax^2 + Bx + C
Math Review November 25, 2014
1. Given
MNPQ ~RSTU
Find the measure
of ∠u. Find the
measure of y.
Similar ~
1. Corresponding
Sides are
proportional.
2. Corresponding
Angles ≅
2. Simplify the
radical. 75
To simplify a radical
means to find
another expression
with the same
value. It
does not mean to find
a decimal
approximation.
3. Graph
y=(x-2)(x+2)
Identify the roots.
Write in standard
form.
Standard
Quadratic
y= Ax^2 + Bx + C
Math Review November 26, 2014
1. Find the
measure of side
lengths: AD and
WX. What is the
similarity ratio
small to large?
Similar ~
1. Corresponding
Sides are
proportional.
2. Corresponding
Angles ≅
2. Simplify the
3. Graph
y=(x-1)(x+1)
radical. 3 50
Identify the roots.
Write in standard
form.
To simplify a radical
means to find another
expression with the
same value. It
does not mean to find
a decimal
approximation.
Standard
Quadratic
y= Ax^2 + Bx + C
Math Review December 1, 2014
1. Find the
2. Simplify the
measure of side
𝑥
radical.
lengths: NM and
𝑥
BC. If similar, what
is the similarity
ratio of large to
small?
To simplify a radical
Similar ~
start by finding
1. Corresponding Sides
factors of perfect
are proportional.
squares.
2. Corresponding
Angles ≅
3. Graph
y=(x-3)(x+3)
Identify the roots.
Write in standard
form.
Standard
Quadratic
y= Ax^2 + Bx + C
Math Review December 2, 2014
1. Find the
similarity ratio 2. Simplify the
𝑥
radical.
small to large.
4+ 𝑥
By which
postulate are
the triangles
similar?
Similar Triangles ~∆
1. AA
2. SSS
3. SAS
To simplify a radical
start by finding
factors of perfect
squares.
3. Graph
y=(x-2)(x+2)
Identify the roots
and vertex.
Write in standard
form.
Standard
Quadratic
y= Ax^2 + Bx + C
Math Review December 4, 2014
1. Find BC and the
similarity ratio
small to large. By
which postulate
are the triangles
similar? What is
the ratio of AS to
A L?
Similar Triangles ~∆
AA, SSS, SAS
2. Simplify the
𝑥
radical.
5− 𝑥
To simplify a
radical start by
finding factors of
perfect squares.
3. Graph
y=(x-1)(x+1)
Identify the roots
and vertex.
Write in standard
form.
Standard
Quadratic
y= Ax^2 + Bx + C
Math Review December 8, 2014
1. Write similarity
2. Simplify the
statement and
5
ratio small to
radical.
2
large. By which
postulate are the
triangles similar?
Similar Triangles ~∆
1. AA
2. SSS
3. SAS
To simplify a radical
start by finding
factors of perfect
squares.
3. Graph
y=(-x-1)(-x-1)
Identify the roots
and vertex.
Write in standard
form.
Standard
Quadratic
y= Ax^2 + Bx + C
Math Review December 9, 2014
1. Find the
similarity ratio 2. Simplify the
radical. 72
large to small.
By which
postulate are
the triangles
similar?
Similar Triangles ~∆
1. AA
2. SSS
3. SAS
To simplify a radical
start by finding
factors of perfect
squares.
3. Graph
y=(x+3)(x+3)
Identify the roots.
Write in standard
form.
Standard
Quadratic
y= Ax^2 + Bx + C
Math Review December 10, 2014
1. Write similarity
2. Simplify the
statement and
radical
ratio large to
small. By which
2 50 − 6 2
postulate are the
triangles similar?
Similar Triangles ~∆
1. AA
2. SSS
3. SAS
To simplify a radical
means to find
another expression
with the same
value. It
does not mean to
find a decimal
approximation.
3. Graph
y=(-x-1)(x-1)
Identify the roots.
Write in standard
form.
Standard
Quadratic
y= Ax^2 + Bx + C
Math Review December 11, 2014
Given
2. Simplify the
∆𝐴𝐵𝐸 ~ ∆𝐶𝐷𝐸.
radical.
Find ?.
𝟑𝟐- 𝟏𝟖
Similar Triangles ~∆
1. AA
2. SSS
3. SAS
To simplify a radical
start by finding
factors of perfect
squares.
3. Graph
y=(-x-1)(x-1)
Identify the roots
and vertex.
Write in standard
form.
Standard
Quadratic
y= Ax^2 + Bx + C
Math Review December 12, 2014
1. Write similarity
statement and
ratio small to
large. By which
postulate are the
triangles similar?
Similar Triangles ~∆
1. AA
2. SSS
3. SAS
2. Simplify the 3. Graph
y=(-x-1)(-x-1)
radical
Identify the roots
108
and vertex.
Write in standard
form.
To simplify a radical
start by finding
factors of perfect
squares.
Standard
Quadratic
y= Ax^2 + Bx + C
Math Review December 13, 2014
1. Find the similarity
ratio large to small.
By which postulate
are the triangles
similar?
2. Simplify
the
radical
26
∠𝐴 = 37°
∠B=91°
Similar Triangles ~∆
1. AA
2. SSS
3. SAS
To simplify a
radical start by
finding factors of
perfect squares.
3. Graph
y=(-x+2)(-x-1)
Identify the roots
and vertex.
Write in standard
form.
Standard
Quadratic
y= Ax^2 + Bx + C