“Ratios and Proportions”

RatioRatio—compares two quantities in a fraction form with one number over another number.

ProportionProportion—two equal ratios.

a

b

a c

b d

To Solve ProportionsTo Solve Proportions“Cross-Multiply”

adad = = bcbcMultiply across from upper

left to lower right and from upper right to lower left.

What do you get?

8x8x = = 5y5y

a

d

c

b 8

5x

y

Ex #1: Solve for x in these Proportions

“Cross-Multiply”5x= 70

x = 14x = 14

“Cross-Multiply”3x = 2(x + 1)3x = 2x + 21x = 2

x = 2x = 2

2

35 5

x

1

2 3

x x

Ex #2: Solve for x in this Proportion

“Cross Multiply”5(x – 7) = 2(x + 5)5x – 35 = 2x + 103x – 35 = 103x = 45

x = 15x = 15

7 5

2 5

x x

STUDY FOR UNIT 4 TEST(Friday , Dec. 14(Friday , Dec. 14thth or Monday, Dec. 17th) or Monday, Dec. 17th)

Triangles Congruent Polygons Congruent Triangles

SSS, SAS, ASA, AAS Problems SSS, SAS, ASA, AAS Proofs CPCTC and CPCTC Proofs

HL Theorem Equilateral, Isosceles, and Right Triangle Problems and

Solving for Missing Variables

Problem of the DayProblem of the DayReturn Test and Collect SOL HW #5

“Angle 1 and Angle 2 are Supplementary Angles. m<1 = 12x + 8 and m<2 = 8x + 12. Find the m<2?”

A.20 degreesB.28 degreesC.104 degreesD.76 degrees

Section 7.2 “Similar Polygons”

Ex I.: Ratio ProblemA Scale Model of a car is 4 in long. The actual

car is 180 in long. What is the ratio of the What is the ratio of the length of the model to the length of the car?length of the model to the length of the car?

Model Car = 4 in Real Car 180 in

1 in1 in45 in45 in

Similar (~) and 1 NoteTwo figures that have the same shape but

not necessarily the same size are similar.

1. When solving Similar Problems, Similar Problems, use ProportionsProportions.

To Tell if Similar Polygons??1. Corresponding Angles

must be Equal.

2. Corresponding Sides must be in Proportional Ratios.

Triangle ABC ~ RSTTriangle ABC ~ RSTB

5050

1010 20 20

85 45 85 45 A 2222 C

1111T 45 85 45 85 R

1010 5050 5 S

Ex #1:Complete each Statement

ABCD ~ EFGHABCD ~ EFGH B C

A 5353 D

F 127 127 G

E H

I. m<E =II. m<B =III. AB = AD

EF ?IV. BC = AB

? EFV. GH = FG

CD ?

Try Example #2I. <HII. <RIII. <XIV. HXV. HR

Ex #3:I. Polygons Similar (Yes/No)? II. Why w/Angles and Sides?

I. B 15

12 ACE 18

16 20

D 24 F

II. Angles Equal B 10

14 ACE 15

18 12

D 20 F

Try Example #4I. No, Sides are Not Proportional RatiosII. Yes, Angles Equal and Sides are ¾ Ratio

Ex #5:For both, Find x and y w/Similar Figures

LMNO ~ QRST T xx

S O 22 N

Q R

1515L 66 M

ABC ~ YWZ Y

4040A 2020

yy x x W Z

B C 3030

1212

Try Example #6I. x = 6 and y = 5II. x = 9

Worksheet “Similar Polygons”

Problem of the Day #1Problem of the Day #1**Check Homework**

1. Yes, by 1/2 Ratio2. No, Angles are not Equal3. x = 44. x = 20 and y = 8

Problem of the Day #2Problem of the Day #2**Show Grades or Show Homework/Return Work**

1. x = 5.3 and y = 362. I. x = 7

II. BC = 11III. MN = 22

“Proving Triangles Similar”

Try—3pts + 2pt EC CorrectFind the Height of the Tree?

2m3m

30m

Example #1“A Small Child is 3 feet tall and is standing 6

feet from a flag pole. Another person is standing 12 feet from the same flag pole. How tall is that other person?”

[Draw Picture]

3 Similarities1. Angle-Angle (AA~) SimilarityAngle-Angle (AA~) Similarity

“If Two Angles Congruent, then Triangles are Similar.”

2. Side-Angle-Side (SAS~) SimilaritySide-Angle-Side (SAS~) Similarity“If Two Sides are Proportional and the middle Angle is Congruent, then Triangles are Similar.”

3. Side-Side-Side (SSS~) SimilaritySide-Side-Side (SSS~) Similarity“If all three Sides are Proportional, then Triangles are Similar.”

One More NoteLittle Triangle Inside Big Triangle, then

AA~AA~.

Ex #2a:Explain why these Triangles are Similar? Write a Similarity Statement?

W T

7070 70 70R M L

<WMR = <TML (Vertical Angles), so

Triangle WMR ~ Triangle TML by AA~AA~

Ex #2b:Explain why these Triangles are Similar? Write a Similarity Statement?

G R 12 M

24 18

F 16 H K<F = <M are Right Angles12/16 = ¾18/24 = ¾

Triangle FGH ~ Triangle MKR by SAS~SAS~

Ex #2c:Explain why these Triangles are Similar? Write a Similarity Statement?

H T10 10 12 12

G C N M

25 3010/12 = 5/610/12 = 5/625/30 = 5/6

Triangle GHC ~ Triangle NTM by SSS~SSS~

Ex #3:Explain why Similar? Then, find x?

A 1010 1212 B C D

1818 X X

E

SSS~ PostulateSSS~ Postulate12 = 1018 X12X = 180

X = 15 X = 15

XX 16 16

44 12 12

SAS~ PostulateSAS~ PostulateX = 412 1616X = 48

X = 3X = 3

Try Examples #4I. AA~ or SAS~ and x = 9II. SAS~ and x = 90

Ex #5:Explain why Similar? Then, find x?

66

99

22

XX

AA~ PostulateAA~ Postulate6 = 6 + 2 89 X6X = 72

X = 12X = 12

44

55

XX

1515

AA~ PostulateAA~ Postulate4 = X + 45 155X + 20 = 60; 5X = 40

X = 8X = 8

EXIT SLIP Worth: +10 Points EXIT SLIP Worth: +10 Points **COLLECT** Explain why Similar? Then, find x? I. A 88 44B C

D 66 X X

E

II.

9090

110110

9090

XX

1. Worksheet “Similarity in Triangles”2. SHORT QUIZ (Next Block)

Similar PolygonsSimilar Polygons Similar Polygon ProblemsSimilar Polygon Problems Similarity in Triangles (AA~, SAS~, Similarity in Triangles (AA~, SAS~, and SSS~)and SSS~)

Similarity in Triangle Solving Similarity in Triangle Solving ProblemsProblems

Problem of the DayProblem of the Day**Check Worksheet and Then Quiz** I. Explain why Similar (AA~, SAS~, SSS~)? II. Find x? 3

7

9

XX

Then New Notes

“Proportions in Triangles”‘‘2’ Theorems2’ Theorems

1. Triangle-Angle-Bisector Theorem

“If an angle bisector bisects a triangle, then it divides the opposite side into two segments that are proportional to the triangle sides.”

A

BC D

AC CD

AB BD

Ex #1:Find x w/Triangle-Angle-Bisector

A 66

X X B D 55

C 88

5x = 48

x = x = 9.6 9.6

8 8

55

xx

33

5x = 24

x = 4.8x = 4.8

8

6 5

xAC CD

AB BD

5 8

3 x

Ex #2:Find x w/Triangle-Angle-Bisector

A

44

66 B D

C xx

66

x = 3.6 x = 3.6

88

66

xx

1010

x = x = 5.75.7

2. Side-Splitter Theorem“If a line is parallel to one side of a triangle and intersects

the other two sides, then it divides those sides in proportions.”

A

c B C a d

bD E

AB AC

BD CE

a c

b d

Ex #3: Find x w/Side-Splitter Theorem

I. T

x 5 x 5

S U 1616 1010R V

Plug in values:

10x = 80

x = 8x = 8

II.

x + 10x + 10 55

xx 33

3x + 30 = 5x30 = 2x

15 = x15 = x

TS TU

SR UV

5

16 10

x

10 5

3

x

x

Try Ex #4:Find x and y?

6 7 x 14

9 y

X = 12 and Y = 10.5X = 12 and Y = 10.5

+ 2pt EC CorrectSolve

“A man who is 6 feet tall casts a shadow that is 4 feet long. At the same time, a nearby flagpole casts a shadow that is 14 feet long. How tall is the flagpole?”

1. Worksheets “2 Theorems”

2. Test—Unit 5 Test—Unit 5 (Friday, January 11(Friday, January 11thth 1 1stst, 5, 5thth, 7, 7thth Blocks Blocks

Monday, January 14Monday, January 14thth 2 2ndnd and 6 and 6thth Blocks) Blocks) Ratio and Proportion ProblemsRatio and Proportion Problems Scale Model w/Ratio ProblemsScale Model w/Ratio Problems Cross-Multiply to Find “x” ProblemsCross-Multiply to Find “x” Problems Similar Polygons and their ProblemsSimilar Polygons and their Problems Similarity in Triangles (AA~, SAS~, and SSS~)Similarity in Triangles (AA~, SAS~, and SSS~) Similarity in Triangle Solving ProblemsSimilarity in Triangle Solving Problems Triangle Angle Bisector and Side-Splitter Triangle Angle Bisector and Side-Splitter

TheoremsTheorems Triangle Inequality ProblemsTriangle Inequality Problems

Problem of the DayProblem of the DayFind x for Both?Find x for Both?

**Check Worksheet**

4 2 X 6

x = 12x = 12

x + 6 8 x – 2

2

X = 4.7X = 4.7

Inequalities in Triangles‘3’ Theorems

1st ‘2’ TheoremsIf an Unequal , (1) the longest side is across

from the largest angle and (2) the largest angle is across from the longest side.

If <A is largest, then BC is the longest side.If <A is largest, then BC is the longest side.

B

C

A

then)

Ex #1:List the Angles in Size from Smallest to LargestList the Angles in Size from Smallest to Largest

K

21 ft21 ft 38 ft38 ft

L 36 ft 36 ft M

Since Side KL is the smallest, <M then <K then <L

(Greater then)

Ex #2:From a Triangle, List Angles from biggest to From a Triangle, List Angles from biggest to smallestsmallestIn Triangle ABC withAB = 12 ftAC = 11 ftBC = 8 ft[Hint: Draw the Triangle]

Biggest to Smallest Angles

<C to <B to <A<C to <B to <A

Ex #3: 1. Find x & 2. Find the Shortest and Longest Sides?1. Find x & 2. Find the Shortest and Longest Sides?

I. N II. H xx x x

6464 C

8282 B 5757 3232

K V

x = 34 degrees x = 91 degreesShortest Side = CV Shortest Side = BHLongest Side = NC Longest Side = BK

(Greater 50

then)(Greater

then)

Ex #4:From a Triangle, List Sides from shortest to longestFrom a Triangle, List Sides from shortest to longestIn Triangle ABC with<A = 90 degrees<B = 40 degrees<C = 50 degrees[Hint: Draw the Triangle]

Short to Long Sides

AC, AB, to BCAC, AB, to BC

33rdrd Theorem Theorem w/Wooden Popsicle Stick Activity (+8pts)w/Wooden Popsicle Stick Activity (+8pts)1. Cut the first Popsicle Stick into 1 in., 1 in., and 4 in. pieces. 2. Try to make a Triangle out of these three cut-out pieces.

Glue them down on a piece of paper.3. Cut the second Popsicle Stick into 2 in., 2 in., and 2 in.

pieces. 4. Try to make a Triangle out of these three cut-out pieces.

Glue them down on a piece of paper.

Answer these Questions: 1. Which makes a Triangle (1st or 2nd Popsicle Stick)? 2. Why??

Inequality Theorem:If the sum of the lengths of a triangle is greater then the third side, then YESYES

a triangle. If not, NONO a triangle.

Ex #5: For all ThreeA Triangle (YES/NOYES/NO)?

I. Sides 3 ft, 7ft, 8ft II. Sides 3cm, 6cm, 1ocm

3 + 6 > 10 (No)3 + 7 > 8 (Yes)7 + 8 > 3 (Yes)3 + 8 > 7 (Yes)

NONO, not a Triangle, not a Triangle

YESYES, a Triangle, a Triangle

III. Sides 1 ft, 9 ft, and 9ft

YESYES, a Triangle, a Triangle

Example #6Which of these three lengths COULD NOT

be the lengths of the sides of a triangle? WHY??

A 7 m, 9 m, 5 mB 3 m, 6 m, 9 mC 5 m, 7 m, 8 m

Try Example #7Which of these three lengths can COULD

be the lengths of the sides of a triangle? WHY??

A 3 m, 14 m, 17 mB 11 m, 8 m, 12 mC 2 m, 3 m, 7 m

Example #8Find the Possible Length of the Third Side, TK?

T35 ft

K 45 ft H

A. 10 < x < 45B. 35 < x < 80C. 10 < x < 80D. 10 < x < 80

(Greater then)

Try Example #9Find the Possible Length of the Third Side, TK?

T25 ft

K 40 ft H

A. 25 < x < 65 B. 15 < x < 40C. 15 < x < 65D. 15 < x < 65

(Greater then)

1. Worksheets “Triangle Inequalities”

2.2. Test—Unit 5 Test—Unit 5 (Friday, January 11(Friday, January 11thth 1 1stst, 5, 5thth, 7, 7thth Blocks BlocksMonday, January 14Monday, January 14thth 2 2ndnd and 6 and 6thth Blocks) Blocks)

Ratio and Proportion ProblemsRatio and Proportion Problems Scale Model w/Ratio ProblemsScale Model w/Ratio Problems Cross-Multiply to Find “x” ProblemsCross-Multiply to Find “x” Problems Similar Polygons and their ProblemsSimilar Polygons and their Problems Similarity in Triangles (AA~, SAS~, and SSS~)Similarity in Triangles (AA~, SAS~, and SSS~) Similarity in Triangle Solving ProblemsSimilarity in Triangle Solving Problems Triangle Angle Bisector and Side-Splitter Triangle Angle Bisector and Side-Splitter

TheoremsTheorems Triangle Inequality ProblemsTriangle Inequality Problems

Problem of the Day**Check HW and Then Test**SOL TEI Review Question

Worth: +10pts1.A2.B3.C4.C5.D6.B7.A8.A9.x = 80 degrees and BD (Longest Side)

120 105

30 130

75 80

1.1. SOL Homework #6SOL Homework #6

2.2. Semester Review BookletSemester Review Booklet

3.3. Extra Credit SheetExtra Credit Sheet