math 9

1. RatioandProportion

3. LearningTargets
Expressratioinsimplestform.
Solvefortheunknownterminagivenproportion.
•Showappreciationoftheapplicationofsimilarity
insolvingreal-lifeproblems.

4. Simplifytheratios:
1.
15
45
2.
15𝑐𝑚
3𝑚
3.
34
74

5. Ratiowrittenin fourways
a:b
atob
a/b
𝑎
𝑏

6. Proportion
Anequationthatequatestworatios.
Eachnumberinaproportioniscalleda
term.Intheproportiona:b=c:d,theterms
aanddarecalledextremeswhilethe
termsbandcarecalledmeans.

7. PropertiesofProportion
1. CrossProductProperty:Theproductpfthe
meansequalstheproductoftheextremes.
2. ReciprocalProperty:Iftworatiosareequal,
thentheirreciprocalsarealsoequal.

8. Findtheunknownintheproportion
8
𝑏
=
𝑏
12
(8)(12) (b)(b)
=
60 = 𝑏2
60 = 𝑏2

9. 8:7=40:x
(8) (x) = (7)(40)
8x = 280

10. 7x-30 : 9x-30 = 5 : 7
(7x-30) (7) = (9x-30)(5)
49x – 210 = 45x - 150
49x – 45x = 210 - 150
4x = 60
x = 15

11. (6)(7) = 2 (y – 3)
42 = 2y - 6
42 + 6 = 2y
48 = 2y
y
=
24

12. (x)(16) = (8)(5)
16x = 40
x = 2.5

13. 4 (x + 3) = 7 (x – 2)
4x + 12 = 7x -14
12 + 14 = 7x -4x
26 3x
=

14. 4 (x + 3) = 7 (x – 2)
4x + 12 = 7x -14
12 + 14 = 7x -4x
26 3x
=

15. Workbook page 79

16. Similarity Between
Triangles

18. LearningTargets
- Illustrate similarities of figure.
- Prove the conditions for similarity of
triangles.
- Determine if two triangles are similar

19. ASA

20. SSS

21. SAA

22. SimilarTriangles
Whenthereisacorrespondingbetween
twotrianglessuchthattheir
correspondinganglesarecongruentand
thelengthsofthecorresponding sides
areproportional.

25. Inthefigure△ABC ∼ △DEF , complete the
following statements given that AB = 6, BC =
8, AC = 10 and DE = 3.
Find the following:
𝐷𝐸
𝐴𝐵
=?
𝑃𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟△DEF
𝑃𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟△ABC
=?

26. ScaleFactor
Theratioofthelengthsoftwocorrespondingsides.

27. Postulate:AAASimilarity
Postulate
Two triangles are similar if and only if
their corresponding angles are
congruent.
Consider △ABC and △EFG . If ∠A ≅
∠E, ∠B ≅ ∠F, and ∠C ≅ ∠G, then we
conclude that △ABC ∼ △EFG .

29. Theorem:AASimilarityTheorem
Iftwoanglesofonetrianglearecongruentto
twoanglesofanothertriangle,thenthe
trianglesaresimilar.

31. Theorem:TheSASSimilarityTheorem
Ifintwotrianglesthetwopairsof
correspondingsidesareproportional
andtheincludedanglesare
congruent,thenthetwotrianglesare
similar.

32. 𝑃𝑄
𝑋𝑌
𝑄𝑅
𝑌𝑍
=
𝑃𝑅
𝑋𝑍
=

33. Determineifthetwogiventrianglesare
similar

34. Determineifthetwogiventrianglesare
similar

35. Theorem:TheSSSSimilarityTheorem
Ifthecorrespondingsidesoftwo
trianglesareproportional,then
thetrianglesaresimilar.

36. 𝐴𝐶
𝐷𝐹
=
10
15
=
2
3
𝐶𝐵
𝐹𝐸
=
8
12
=
2
3
𝐴𝐵
𝐷𝐸
=
12
18
=
2
3

37. 𝐴𝐶
𝐷𝐸
=
8
6
=
4
3
𝐶𝐵
𝐹𝐸
=
16
12
=
4
3
𝐴𝐵
𝐷𝐸
=
12
9
=
4
3

38. ACTIVITY

39. ACTIVITY

40. ACTIVITY

41. Similarities in Right
Triangles

42. LearningTargets

45. Other Theorems on Right
Triangles

46. LearningTargets
StateandprovethePythagorean
Theorem;
Stateandprovethetheoremsonspecial
righttriangle;and
Solveproblemsthatinvolvetriangle
similarityandrighttriangles.

47. PythagoreanTheorem
Inarighttriangle, thesquareofthehypotenuse isequalto thesum ofthesquares of the
legs.