Name: ______ Period _________ 2/17/12 – 3/2/12 G-PreAP

UUNITNIT 12: S 12: SIMILARITYIMILARITY

I can define, identify and illustrate the following terms: DilationScale FactorExtremesMeans

Similarity StatementScale DrawingEnlargementReduction

RatioProportionCross productsIndirect measurement

Similarity ratio

Name:____________________________Period:______ 1/20-21/2010 GH

Dilations as Proportions

Dates, assignments, and quizzes subject to change without advance notice.

Monday Tuesday Block Day Friday13

12- 4 and 12 – 5Compositions and

Symmetry

1412-7

Dilations

15/16ReviewTTESTEST 11 11

177-1

Ratios, Proportions, and Problem Solving

20

No School

217-2

Similar Polygons

22/237-3 & 7-4

Triangle Similarity & Applications

QUIZ 1

247-5

Proportional Relationships

27 7-6

Dilations & Similarity in the Coordinate Plane

288-1

Geometric MeanQUIZ 2

29/1ReviewTTESTEST 12 12

2

Friday, 2/17

7-1: Ratio and Proportion

I can write a ratio. I can write a ratio expressing the slope of a line. I can solve a linear proportion. I can solve a quadratic proportion.

PRACTICE: Pg 458 #8-13, 17-29

Tuesday, 2/21

7-2: Similar Polygons

I can use the definition of similar polygons to determine if two polygons are similar. I can determine the similarity ratio between two polygons.

PRACTICE: Pg 465 #7-20, 27-30 Ratio Problems Worksheet

Wednesday, 2/22 or Thursday, 2/23 QUIZ 1: 7-1 & 7-2

7-3: Triangle Similarity & 7-4: Applications and Problem Solving

I can use the triangle similarity theorems to determine if two triangles are similar. I can use proportions in similar triangles to solve for missing sides.

I can use the triangle proportionality theorem and its converse. I can use the Triangle Angle Bisector Theorem. I can set up and solve problems using properties of similar triangles.

PRACTICE: Pg 475 #11-14, 16, 20-24 Pg 485 #8-20, 25, 28, 32, 34

Friday, 2/24

7-5: Using Proportional Relationships

I can use proportions to determine if a figure has been dilated. I can use ratios to make indirect measurements.

PRACTICE: Pg 491 #18-19, 24-26, 28-29, 34 Proportional Relationships Worksheet

Monday, 2/27

7-6: Dilations & Similarity in the Coordinate Plane

I can use coordinate proof to prove figures similar. I can apply similarity in the coordinate plan.

PRACTICE: Pg 498 #4-7, 11-14, 21-24

Tuesday, 2/28 QUIZ 2: Similar Triangles (7-3 through 7-6)

Geometric Mean

I can use geometric mean.PRACTICE: Geometric Mean Worksheet

Wednesday, 2/29 or Thursday, 3/1

Review

Test 12: Similarity I can use ratios and proportions to show figures are similar or solve problems with similar figures.

PRACTICE: Review Worksheet; Pg 504 – 507 has even MORE practice if you want it.

Dilations as Proportions Notes

Ex) Rectangle CUTE was dilated to create rectangle UGLY. Find the length of LY.

C U

TE

8 cm3 cm

U G

LY7.5 cm

Ex) Determine which of the following figures could be a dilation of the triangle to the right. (There could be more than one answer)

1. Find the length of after the dilation.

2. Which of the following could NOT be an enlargement or reduction (dilation) of the original painting shown at right?

A B

C D

16 in.

6 in. 6 in.

2.25 in.

20 in.

10 in.

8 in.

3 in.

30 in.

5 in.

A B C D

C′

A

C

A′

BB′

6 m10 m4 m

12 in

8 in

13 in

9in

18 in

12 in

15 in

10 in

6 in

4 in

Ratio ProblemsWrite the equation for each and solve. Show all work.

1. One angle of a triangle measures 18º. The other two angles are in the ratio of 4:5. Find the measures of the angles.

2. The vertex angle and the one base angle of an isosceles triangle are in the ratio of 7:4. Find the measures of the 3 angles.

3. The sides of a triangle have the ratio of 7:9:12. The perimeter of the triangle is 420. Find the length of each side.

4. Two consecutive angles of a parallelogram are in a ratio of 7:3. Find the measures of the angles.

5. The angles of a pentagon are in a ratio of 7:6:5:5:4. Find the measures of each angle.

6. One angle of a hexagon is 120º. The other angles are in the ratio of 11:9:8:7:5. Find the measures of the angles.

7. The sides of a quadrilateral are 5, 7, 4, and 8 cm. If the shortest side of a similar quadrilateral is 10cm, what is the length of the longest side?

8. If a rectangular I.D. card is 4cm x 8cm (l x w) is enlarged so that its new perimeter is 50cm, what is the new width?

9. The base of an isosceles triangle is 12cm long and one of the legs is 6cm long. What is the perimeter of a similar triangle whose base is 18cm?

10. The sides of a polygon are 3, 4, 7, 9, and 11 mm long. Find the perimeter of a similar polygon whose shortest side is 10.5 mm long.

Notes: Similar Triangles

There are 3 ways you can prove triangles similar WITHOUT having to use all sides and angles.

Angle- Angle Similarity (AA~) – If two angle of one triangle are ______________ to two corresponding

angles of another triangle, then the triangles are similar

Side- Side- Side Similarity (SSS~) – If the three sides of one triangle are __________________ to the

three corresponding sides of another triangle, then the triangles are similar.

Side-Angle- Side Similarity (SAS~) – If two sides of one triangle are ____________________ to two

corresponding sides of another triangle and their included angles are ________________, then the triangles

are similar.

Examples: Determine if the triangles are similar. If so, tell why and write the similarity statement and similarity ratio.

Similar : Y or N Why:_________ Similar : Y or N Why:_________

Similarity Statement :_______~__________ Similarity Statement :_______~__________

Similarity Ratio :__________ Similarity Ratio :__________

Similar : Y or N Why:_________ Similar : Y or N Why:_________

Similarity Statement :_______~__________ Similarity Statement :_______~__________

Similarity Ratio :__________ Similarity Ratio :__________

Notes: Properties of Similar Triangles

Ex. 1 Ex. 2

Now that you can write the proportions, you can solve problems.

Ex. 3 Ex. 4 Ex. 5

Ex. 6 Find RV Ex. 7 Find y

V

I

D

E

OP

A

Y

E

SL R

V

I

D

E

O

x

5

8

20x x

Proportional Relationships

Solve for the variable in each figure.

1. y =__________ 2. k =______________ x = __________

3. SQ = x ; ST = 22 ; 4. y =__________ SP = 12 ; PR = 4x+8

x = __________ x = __________

Given:

5. AC = 9; BC = 6; DF = 15

EF = __________

6. AB = 5y; DE = 2y; EF = 12

BC = ___________

7. BC = x + 2; BA = 9; EF = x + 3; ED = 12

x = _______; BC = ________; EF = _______

8. AC = 3x; BC = 16; EF = 20; FD = 4x -2

5

23

5y – 2

3x + 2

x

10

k

923 4

12

8

y3

x

5

20

16

j

k

l

A

B

C

D

E

F

x = ______; AC = _______; FD = _______

Dilations & Similarity in the Coordinate Plane Notes

5. 6.

7.

10.

11. Given: E(–2, –6), F(–3, –2), G(2, –2), H(–4, 2), and J(6, 2). Prove: ∆EHJ ~ ∆EFG.

12. Given: R(–2, 0), S(–3, 1), T(0, 1), U(–5, 3), and V(4, 3). Prove: ∆RST ~ ∆RUV

8. 9.

Notes: Geometric Mean

The geometric mean of two positive numbers is the positive square root of their products. For , the x

is the geometric mean.

Examples: Find the geometric mean of the given numbers.

a. 4 and 9 b. 6 and 15 c. 2 and 8

The altitude to the hypotenuse of a right triangle forms two triangles that are ________________ to each

other and to the original triangle. In other words, there are ________ similar triangles: a small one, a

medium one, and a large one.

Examples:

Ex 1: m = 2, n = 10, h = _____

Ex 2: m = 2, n = 10, b = _____

c

a b

nm

h

a

b

ch b

n

m

h

a

c

a b

nm

h

You draw the similar triangles:

Geometric Mean

If ∆ABC is a right triangle and is the altitude to the hypotenuse then

∆ABC ~ ∆CBD → →a² = mc

∆ABC ~ ∆ACD → → b² = nc

∆ACD ~ ∆CBD → → h² = mn

1. c = 12; m = 6; a = ? 2. m = 4; h = 25; n = ? 3. c = 12; m = 4; h = ?

4. a = 30; c = 50; h = ? 5. h = 12; m = 9; b = ? 6. a = 24; m = 4; b = ?

7. b = 45; n = 5; a = ? 8. b = 8; m = 12; c =?

c

a b

nm

h

A BD

C

a

b

ch b

n

m

h

a

10. a = ; h = 14; c = ?; n = ? 11. a = ; b = ; m = ?; h = ?