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NAME _____________________________________ COMMON CORE GEOMETRY

Module 2 Part II

Setting up proportions with similar triangles

And Simplifying Radicals

DATE PAGE TOPIC HOMEWORK

12/11 2-3 Lesson 1: Similar triangles and proportions Start Lesson 2

Homework Worksheet

12/12 4-5 LESSON 2: Applying the Triangle Side Splitter Theorem

Homework Worksheet

12/15 6-7 LESSON 3: Applying the triangle side splitter theorem continued

Homework Worksheet

12/16 8-10 Lesson 4: Ratios of Sides, Perimeters, and Areas Homework Worksheet

12/17 11-12 QUIZ Lesson 5: The Angle Bisector Theorem

Homework Worksheet

12/18 13-15 Lesson 6: Special Relationships Within Right Triangles—Dividing into Two Similar Sub-Triangles

Homework Worksheet

12/19 16-17 Lesson 7: Special Relationships Within Right Triangles- Another useful proportion

Homework Worksheet

12/22 18-19 QUIZ LESSON 8: A side note- what to do if we get a radical?

No Homework

12/23 20-22 Lesson 9: More operations with radicals No Homework

1/5/15 23-24 LESSON 10: Adding and Subtracting Radicals Homework Worksheet

1/6 25-26 LESSON 11: Putting It all Together Homework Worksheet

1/7 QUIZ Review

Finish Review Packet

1/8 Review Study

1/9 Test No Homework

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Lesson 1: Similar triangles and proportions

Prove triangles are similar:

a.) In the diagram below DE is parallel to AB, mark your picture accordingly:

b.) Fill in the appropriate givens and what you are trying to prove-

Given: Prove:

c.) Proof:

d.) Draw the similar triangles separately and label:

D.)Now that we know that the triangles are similar, Let’s fill in the following proportions:

A

D

C

E

B

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EXERCISES: In 1-3 find x given that ̅̅ ̅̅ ̅̅ ̅̅

1. 2.

3.

4. A vertical pole, 15 feet high, casts a shadow 12 feet long. At the same time, a nearby tree casts a shadow 40

feet long. What is the height of the tree?

5. Caterina’s boat has come untied and floated away on the lake. She is standing atop a cliff that is 35 feet

above the water in a lake. If she stands 10 feet from the edge of the cliff, she can visually align the top of the

cliff with the water at the back of her boat. Her eye level is 5.5 feet above the ground. How far out from the

cliff, to the nearest tenth, is Catarina’s boat?

35ft

10ft

5.5ft

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LESSON 2: Applying the Triangle Side Splitter Theorem

Side Splitter Theorem:

A line segment splits two sides of a triangle proportionally if and only if it is parallel to the third side.

Using this Theorem, answer the following questions:

1. If ̅̅ ̅̅ ̅̅ ̅̅ , ̅̅ ̅̅ , ̅̅ ̅̅ , and ̅̅ ̅̅ , what is ̅̅ ̅̅ ?

2. If ̅̅ ̅̅ ̅̅ ̅̅ , ̅̅ ̅̅ , ̅̅ ̅̅ , and ̅̅ ̅̅ , what is ̅̅̅̅ ?

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6. In the diagram pictured, a large flag pole stands outside of an office building. Josh realizes that when he

looks up from the ground, 60m away from the flagpole, that the top of the flagpole and the top of the building

line up. If the flagpole is 35m tall, and Josh is 170m from the building, how tall is the building to the nearest

tenth?

4. 5.

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LESSON 3: Applying the triangle side splitter theorem continued

Opening Exercise

In the two triangles pictured below, ̅̅ ̅̅ ̅̅ ̅̅ and ̅̅ ̅̅ ̅. Find the measure of x in both triangles

What is the relationship between the triangles ABC and FGH?

Since the two triangles share a common side, look what happens when we push them together:

Now we have three parallel lines cut by transversals, is the transversal on the left in proportion to the

transversal on the right?

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Exercises

In exercises 1 and 2, find the value of x. Lines that appear to be parallel are in fact parallel

1.)

2.)

3.) In the diagram below, ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ , AB=20, CD=8, FD=12 and AE:EC=1:3. If the perimeter of the

trapezoid ABCD is 64, find AE and EC.

THEOREM: If 3 or more lines are cut by 2 transversals, then the segments of the transversals are in

proportion.

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Lesson 4: Ratios of Sides, Perimeters, and Areas

Exercise

Given ABCA’B’C’ pictured to the right:

a. Find the lengths of the missing sides.

b. Find the perimeters of the triangles.

c. Find the areas of the triangles.

Area formula:

d. What is the ratio of the sides of the triangles?

e. What is the ratio of the perimeters of the triangles?

f. What is the ratio of the areas of the triangles?

RULES:

The ratio of the perimeters is _____________________________________________________

The ratio of the areas is _________________________________________________________

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Example 2

Let’s test the hypothesis we made in Exercise 1.

Given the similar triangles pictured to the right, find:

a.) The ratio of the sides

b.) The ratio of the perimeters

c.) The ratio of the areas

Given the similar rectangles pictured below, find:

a.) The ratio of the sides

b.) The ratio of the perimeters

c.) The ratio of the areas

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Examples:

1.) When two figures are similar and the ratio of their sides is a:b, then:

The ratio of their perimeters is:

The ratio of their areas is:

2.) Two triangles are similar. The sides of the smaller triangle are 6,4,8. If the shortest side of the larger

triangle is 6, find the length of the longest side.

3.) The sides of a triangle are 8, 5, and 7. If the longest side of a similar triangle measures 24, find the

perimeter of the larger triangle.

4.) Find the ratio of the lengths of a pair of corresponding sides in two similar polygons if the ratio of the

areas is 4:25.

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Lesson 5: The Angle Bisector Theorem

The angle bisector theorem states:

Exercises:

1.) In ABC pictured below, AD is the angle bisector of A. If CD=6, CA=8 and AB=12, find BD.

2.) In ABC pictured below, AD is the angle bisector of A. If CD=9. CA=12 and AB=16. Find BD.

𝐵𝐷

𝐶𝐷 𝐵𝐴

𝐶𝐴

In ABC, if the angle bisector of A

meets side BC at point D, then

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3.) The sides of ABC pictured below are 10.5, 16.5 and 9. An angle bisector meets the side length of 9. Find

the lengths of x and y.

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Lesson 6: Special Relationships Within Right Triangles—Dividing into Two Similar Sub-Triangles

Opening Exercise

Use the diagram to complete parts (a)–(c).

a. Are the triangles shown similar? Explain.

b. Determine the unknown lengths of the triangles using Pythagorean Theorem.

Example 1

In ABC pictured to the right, B is a right angle and BC is the altitude.

DEFINE: Altitude: ____________________________________________________________________

a. How many triangles do you see in the figure? Draw them:

b. In the altitude ̅̅ ̅̅ divides the right triangle into two sub-triangles, and

Is ?

Is ?

Since and can we conclude that ? Explain.

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Example 2

Consider the right triangle below.

a. Altitude ̅̅ ̅̅ is drawn from vertex to the line containing ̅̅ ̅̅ Segment ̅̅ ̅̅ , segment ̅̅ ̅̅ ,

and the segment ̅̅ ̅̅ as

b. Draw the two similar triangles ADB and BDC and label the sides appropriately.

c. Set a proportion to find the value of

d. We should notice that the altitude of the whole triangle is the long leg in the small triangle and is the

short leg in the medium triangle. Therefore we can really use a short cut when we see a diagram like

this:

4 x

8

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EXAMPLES: 1.) 2.)

3.) Given triangle with altitude ̅, find , .

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Lesson 7: Special Relationships Within Right Triangles- Another useful proportion

Consider the right triangle below (same as we worked with yesterday).

a. Altitude ̅̅ ̅̅ is drawn from vertex to the line containing ̅̅ ̅̅ Segment ̅̅ ̅̅ , segment ̅̅ ̅̅ ,

and the segment ̅̅ ̅̅ as

b. Draw the two similar triangles ADB and ABC and label the sides appropriately.

c. Set a proportion to find the value of

d. We should notice that the hypotenuse of the small triangle is the leg in the large triangle and they

hypotenuse in the large triangle is the leg in the short triangle. Therefore we can really use a short cut

when we see a diagram like this:

4

20

x

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EXAMPLES:

1.) 2.)

what is the length of ? What is the length of ?

3.) In the diagram below, the length of the legs and of right triangle ABC are 6 cm and 8 cm,

respectively. Altitude is drawn to the hypotenuse of . What is the length of to the nearest tenth of

a centimeter?

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LESSON 8: A side note- what to do if we get a radical? Opening Exercise Solve for x: Consider our answer, what can we do with it? If we round we are not using the most accurate answer so we want to leave it in simplest radical form. (we’ll come back and do this later) Perfect Squares- ________________________________________________________________________ Identify some perfect squares below:

Factors: Perfect square

12

22

33

42

52

62

72

82

92

102

112

*make a note about this:

Factors: Perfect square

xx

x2x2

X3x3

X4x4

X5x5

X6x6

X7x7

6 8

x

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Write the square roots of the following:

1. √ 2. √ 3. √

4. √ 5. √

Simplify:

6. √ 7. √ 8. √

9. √ 10. √ 11.) go back and simplify

your answer from the

opening exercise.

√𝑛𝑜𝑛 − 𝑝𝑒𝑟𝑓𝑒𝑐𝑡 𝑠𝑞𝑢𝑟𝑒 √𝐿𝑎𝑟𝑔𝑒𝑠𝑡 𝑝𝑒𝑟𝑓𝑒𝑐𝑡 𝑠𝑞𝑢𝑎𝑟𝑒 𝑓𝑎𝑐𝑡𝑜𝑟 ∗ √𝑟𝑒𝑚𝑎𝑖𝑛𝑖𝑛𝑔 𝑓𝑎𝑐𝑡𝑜𝑟

Simplifying Non-Perfect squares:

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Lesson 9: More operations with radicals

1. Complete parts (a) through (c).

a. Compare the value of √ to the value of √ √

b. Make a conjecture about the validity of the following statement. For nonnegative real numbers and

, √ √ √ . Explain.

c. Does your conjecture hold true for and ?

2. Complete parts (a) through (c).

a. Compare the value of √

to the value of

√

√ .

b. Make a conjecture about the validity of the following statement. For nonnegative real numbers and

, when , √

√

√ . Explain.

c. Does your conjecture hold true for and ?

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Exercises 3–12

Simplify each expression as much as possible and rationalize denominators when applicable.

3. √ √

4. √

5. √ √

6. √

7. √ √

8. √

√

9. √

10.

√

11.

√

12. √

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13. Find the area of the figure below:

14. Calculate the area of the triangle:

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LESSON 10: Adding and Subtracting Radicals

1.)

a. Calculate the perimeter of the triangle below:

b. Calculate the perimeter of the triangle:

Since the radicals are not the same we need to do some work before we can add.

*Simplify each side then add.

√ √

√

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SIMPLIFY

2.) √ √ 3.) √ √

4.) √ √ . 5.) √ √ − √

Find the Perimeters

6.) 7.)

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LESSON 11: Putting It all Together

Opening Exercise

In the diagram below of right triangle ABC, an altitude is drawn to the hypotenuse .

Recall the proportions that we set up back in lessons 6 and 7 and fill in the appropriate proportions below:

𝒙

𝒛

𝒄

𝒃

𝒄

𝒂

1.) Triangle ABC shown below is a right triangle with altitude drawn to the hypotenuse .

If and .

Write all answers in simplest radical form:

a.) what is the length of AD? C.) what is the length of AC?

b.) What is the length of AB? D.) What is the area of triangle ABC?

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2.) Given the diagram below, find:

a.) Length of AB

b.) Length of AD

c.) Length of AC

D.) Perimeter of triangle ABC.

E.) Express the area of triangle ABC in simplest form:

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