Download Similarity - Frost Middle School

Similarity
Lesson 12-3
15. XZ ≈ 3.34
(pp. 731–737)
Mental Math
3
a. 12 ft
16. 93 in.
17. a. 200 mi
1
b. _______
2,534,400
b. 3
18. about 17.14 ft
c. 16
Guided Example 1
a. similarity; a size transformation; a reflection
b. m∠H
c. 1.8; 6; 1.8; 10.8
19. Yes, the rectangles must be similar because
corresponding angles are congruent (all angles
are right angles) and corresponding lengths are
3 cm
2 cm = ____
proportional _____
.
( 4 ft
)
20. B'C' = 32
21. x = –5
Questions
1. a. The letters are similar because the d is the
image of the b under a reflection and size
transformation.
22. x - __3
(
2
) + ( y - __72 )
2
2
29
= __
2
23. 96.28% air, 3.72% leather
24. 8; rectangle, rhombus, isosceles trapezoid,
square, parallelogram .
b. __3
4
c. __4
3
2. Two figures are similar if and only if there is a
composite of size transformations and reflections
mapping one onto the other.
3. a. false
b. true
4. Yes, the figures are similar because
LMNK = S2.3 rm(ABED).
5. GF = 6.8 in.; LK = 15.64 in.
6. HJ = 100 mm; DE = 100 mm
7. ∠ABE and ∠LKN
8. CM = 9.2
9. yes
10. yes
11. No; Figure I and Figure III in Question 4 are
similar, but AB ≠ LK.
12. No; Figure I and Figure II in Question 4 are
similar, but oppositely oriented.
13. PA = 4.8
14. m∠O = 80; m∠A = 119
A221
6 ft
Geometry
25. true
26. There are comparable properties for similarity.
Proof of the Reflexive Property of Similarity:
Let F be any figure. We need to show that F is
similar to itself. To do that, reflect F over the line
m, apply size transformation S with center C and
magnitude k, then apply size transformation S' with
center C and magnitude __1 , and reflect that image
k
back over m. There is a composite of reflections
and size transformations mapping F onto itself.
Thus by the definition of similarity, F ∼ F.
Proof of the Symmetric Property of Similarity:
If F ∼ G then there is a composite of reflections
and size transformations that maps F onto G.
To map G back onto F, reflect G back over the
same lines and apply size transformations with
reciprocal magnitudes in reverse order. This
means F is the image of G under a composite of
reflections and size transformations, so G ∼ F.
Proof of the Transitive Property of Similarity:
If F ∼ G then there is a composite of reflections
and size transformations that maps F onto G. If
G ∼ H then there is a composite of reflections
and size transformations that maps G onto H.
The composite of these composites is another
composite of reflections and size transformations
which maps F onto H, so F ∼ H.
A222
Geometry