Download 11_2 Dilations - Similar Triangles

Dilation Similarity Theorem:
If one polygon is a dilated image of another polygon, then
the polygons are similar.
A dilation is a type of transformation that changes the size of an
image based on a central point and a scale factor.
 A dilation takes a line not passing through the
center of the dilation to a parallel line, and
leaves a line passing through the center
unchanged.
 Dilation means choosing a point to
be the center, and drawing lines
that go through the center and a
vertex of the object. To make an
object that is similar to the original
object, you can draw lines that are
parallel to the lines of the object.
The intersections with the lines that
go through the center are then the
points of the new similar object.
 Scale Factor=1, the figures are
congruent.
 Scale factor between 0 and 1, the image
is smaller.
 Scale factor > 1, the image is bigger.
Dilating a line that does not pass through the center of dilation
d. What is the relation between the original line and its dilated image ?
Solution:
d. When dilating, a line that does not pass through the center of
dilation, is parallel to the original line.
Example
Y’(6,15)
Y(2,5)
R(2,2)
R’(6,6)
O(6,2)
O’(18,6)
Where (a,b) are the x and y coordinates of
the center of dilation; and k is the scale factor
Of the dilation.
As a result, when the center of dilation is the origin (0,0),
then:
𝑫
𝟎,𝟎 ,𝒌
𝑫
𝒙, 𝒚 = (𝟎 + 𝒌 𝒙 − 𝟎 , 𝟎 + 𝒌(𝒚 − 𝟎)
𝟎,𝟎 ,𝒌
𝒙, 𝒚 = (𝒌𝒙, 𝒌𝒚)
11.2: Similar Triangles:
3 Similarity shortcuts
1. AA Similarity shortcut: If two angles of one triangle are
congruent to two angles of another triangle, then the
triangles are similar.
Example:
Are triangles A & B similar?
By the triangle sum theorem
57°+ 83° + x = 180°
X= 40°
x
A
B
Yes,
By AA similarity theorem
2. SSS Similarity shortcut: If the three sides of one
triangle are proportional to the three sides of
another triangle, then the two triangles are similar.
Example:
Are Triangles # 1 and # 2 similar?
Yes,
𝟔
𝟖
𝟏𝟎
#1
#2
=
=
𝟑
𝟒 By SSS similarity theorem
𝟓
3. SAS Similarity shortcut: If two sides of one triangle are
proportional to two sides of another triangle and the
included angles are congruent, then the two triangles are
similar.
Example:
Are the triangles below similar?
𝟏𝟐 = 𝟏𝟓
𝟑𝟔
𝟒𝟓
Yes, they are similar,
by SAS similarity theorem
In summary, there are 3 similarity shortcuts
to prove that two triangles are similar:
1. AA similarity shortcut
2. SSS similarity shortcut
3. SAS similarity shortcut
Homework
Workbook 11.2
Except # 8
Classwork
Work with a partner
Hint: Use Pythagorean Theorem
a² + b² = c²
Hint: Use proportions, then use
the Pythagorean Theorem
a² + b² = c²
No, not all isosceles triangles are similar
No, not all right triangles are similar
Yes. All isosceles right triangles are similar