Download 3.1.1 notes

3.1.1 NOTES
Triangle Similarity
WARMUP
• Solve for “x” in the following proportions
x 3
1.
=
6 9
x+4 4
3.
=
5
10
8 7
2.
=
x 24
PRETEND…
• You've been caught up in a twister that deposits you
and your little dog in the middle of a strange new
land. In order to get home, you must prove that two
triangles are similar. How will you do it?
• Just click your heels together three times and say,
"There's no place like my Math 3 classroom.
There's no place like my Math 3 classroom.
There's no place like my Math 3 classroom."
SIMILAR TRIANGLES
Two triangles are similar if they have:
• All their angles equal
• Corresponding sides are in the same ratio
We don’t need to know all six pieces…sometimes
knowing 2 or 3 pieces of information about the triangle
is enough to determine similar
AA (Angle-Angle)
• If two triangles have two of their angles equal, then the
triangles are similar
• Example: these two triangles are similar
• If two of their angles are equal, then the third must
also be equal because angles of a triangle always
make up 180 degrees.
SSS (Side-Side-Side)
• If two triangles have three pairs of sides in the same ratio, then
the triangles are similar.
• Example: these two triangles are similar
• AB : XY = 4 : 5
BC : YZ = 6 : 7.5
AC : XZ = 8 : 10
• These ratios are all equal, so the triangles are similar.
SAS (Side-Angle-Side)
**A lil mashup of AA and SSS**
• If two triangles have two pairs of sides in the same ratio and
the included angles are also equal, then the triangles are
similar
• AB : YX = 15 : 10
• AC : XZ = 21 : 14
• Angle A = Angle X, which is included between the two
sides
TRIANGLE MIDSEGMENT
THEOREM
Sounds like fun, yes?
• If you connect the midpoints of two sides of a triangle, then
you've got yourself a midsegment, a magical creature
that lives smack dab in the middle of the triangle it calls
home. They are:
• half the length of the side they run parallel to
• bisect the other two sides
• B is the midpoint of AC
• D is the midpoint of CE
• We can be sure of that because we're told
that the segments are congruent on each
side of both of those points. Connecting
them, we get the midsegment BD. TMT says
that BD || AE and that BD = 1/2 × AE
TRIANGLE WITH // LINE
• If a parallel line runs through two segments on a triangle, then
the segments it splits are proportional
• Note: This is different from Triangle Midsegment
• You can create two similar triangles from a “parallel side
splitter
WARMUP:
1) Suppose ΔABC ~ ΔEFG and AB =10, AC = 6, EG
= 4, FG = 2.
a) Find BC
a) Find EF
2) Find BC
Find AC
6
10
GEOMETRIC MEAN
• The Geometric Mean is a special type of average
where we multiply the numbers together and then
take a square root (for two numbers), cube root (for
three numbers) etc.
• TRIANGLE WITH ALTITUDE:
• The value that is repeated in a proportion is the
geometric mean
• EX:
2
x
x
8
TRIANGLE WITH
ALTITUDE
• Creates THREE similar right
triangles, a small, medium, and
large:
• Proportions can be made from the 3 triangles:
• The legs of the big triangle become the hypotenuse of the small and
medium
• The hypotenuse of the big triangle splits up, each becoming a leg in
the small and medium
• The altitude becomes a leg in both the small and the medium triangle
Ex 1
x
4
5
Ex 2
x
2
6
FOOTBALL LINE UP
Ex 3
4
x
3
Ex 4
x
12
5