Download Unit 10 Similarity Notes

Unit 10: Similarity
(7-2) Similar Polygons
Similar figures –
Similar Polygons – two polygons are similar polygons if corresponding angles are congruent
and if the lengths of corresponding sides are proportional
ABCD ~ GHIJ
Diagram:
Scale factor –
Understanding Similarity
Example 1:
∆MNP ~ ∆SRT
a) Congruent angles?
b) Extended proportion for the
ratios of corresponding sides?
Unit 10: Similarity
Example 2:
DEFG ~ HJKL
a) Congruent angles?
b) Extended proportion for ratios?
Determining Similarity
Example 3:
Are the polygons similar?
JK
=
TU
KL
=
UV
LM
VW
JM
TW
Example 4:
=
=
Are the polygons similar?
AB
DE
BC
EF
AC
DF
=
=
=
Unit 10: Similarity
Using Similar Polygons
Example 5:
ABCD ~ EFGD. What is the value of x?
What is the value of y?
Scale Drawing
Scale drawing –
Scale – the ratio that compares each length in the scale drawing to the actual length/
Example 6:
The length of a bridge in a drawing is 6.4 cm. The scale drawing is 1 cm = 200 m.
Extra Example: Problem 4, Page 443
Unit 10: Similarity
(7-3) Proving Triangles Similar
 Angle – Angle Similarity (AA ~ ) Postulate
If two angles from one triangle are congruent to two angles of another, then triangles are similar
∆SRT ~ ∆MLP
Diagram:
Example 1:
Are the triangles similar?
Example 2:
Are the triangles similar?
Example 3:
Are the triangles similar?
Unit 10: Similarity
 Side-Angle-Side Similarity (SAS ~ ) Theorem
Contains one pair of congruent angles; Sides that include the two angles are proportional
Diagram:
If…
AB
QR
=
AC
QS
and ∠A ≅ ∠Q
Then… ∆ABC ~ ∆QRS
Example 4:
Are the triangles in this figure similar?
 Side-Side-Side Similarity (SSS ~ ) Theorem
Corresponding sides are proportional
Diagram:
If…
AB
QR
=
AC
QS
=
BC
RS
Then… ∆ABC ~ ∆QRS
Unit 10: Similarity
Example 5:
Are the triangles similar?
Shortest sides:
Longest sides:
ST
XV
US
WX
Remaining sides:
=
=
TU
VW
=
 Indirect Measurement
- Uses the fact that light reflects off a mirror at the same angle at which it hits the
mirror
Example 6: Before rock climbing, Darius wants to know how high he will climb. He places a
mirror on the ground and walks backward until he can see the top of the cliff in the mirror. What
is the height of the cliff?
Unit 10: Similarity
(7-4) Similarity in Right Triangles
 Theorem 7-3
The altitude to the hypotenuse of a right triangle divides the triangle into 2 similar triangles
that are also similar to the original
Example 1:
Write the similarity statement relating the three triangles.
 Geometric Mean
For any two positive numbers a and b, the geometric mean of a and b is the positive number x
a
x
such that
=
x
b
Example 2:
What is the geometric mean of 6 and 15?
Example 3:
What is the geometric mean of 4 and 18?
Unit 10: Similarity
 Corollary 1 to Theorem 7-3
and
Corollary 2 to Theorem 7-3
Unit 10: Similarity
(Concept Byte) The Golden Ratio – Page 468
Unit 10: Similarity
(7-5) Proportions in Triangles
 Side-Splitter Theorem
If
⃡
RS
Then…
Example 1:
What is the value of x?
Example 2:
What is the value of a?
∥
⃡
XY
XR
RQ
=
YS
SQ
Unit 10: Similarity
 Corollary to Side-Splitter Theorem
If a
∥b∥c
Then…
Example 3:
AB
BC
=
WX
XY
Problem 2, Page 473: Three campsites are shown. What is the length of Site A?
Diagram:
Let x be the length of Site A.
Example 4:
What is the length of Site C?
Let y be the length of Site C.
Unit 10: Similarity
 Triangle-Angle-Bisector Theorem
If AD bisects ∠ CAB
Then…
Example 5:
What is the value of x?
Example 6:
What is the value of y?
CD
DB
=
CA
BA