honors math 2 unit 6 class packet sanderson high …...1 day 1: pythagorean theorem and converse...

Honors Math 2 Unit 6 Class Packet Sanderson High School

1

Day 1: Pythagorean Theorem and Converse

Warm-Up:

1. Find the length of the missing side. The triangle is not drawn to scale.

2. Find the length of the missing side. Leave your answer in simplest radical form.

3. A triangle has sides of lengths 12, 14, and 19. Is it a right triangle? Explain.


2

I. Classifying Triangles by their angles

a. Acute Triangle

i. An acute triangle is a triangle that has ___________________________.

b. Obtuse Triangle

i. An obtuse triangle is a triangle that has __________________________.

c. Right Triangle

i. A right triangle is a triangle that has ____________________________.

d. Oblique Triangle

i. An oblique triangle is a _______________________________________.

ii. These can be _______________ triangles or _____________ triangles

e. Equiangular Triangle

i. An equiangular triangle is a triangle that has _______________________.

II. Classifying Triangles by their sides

a. Scalene Triangle

i. A scalene triangle is a triangle that ______________________________.

b. Isosceles Triangle

i. An isosceles triangle is a triangle that has _________________________.

c. Equilateral Triangle

i. An equilateral triangle is a triangle that has _______________________.


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III. Right Triangles and Special Sides

a. A right triangle has three special sides

b. These sides are dependent on the angles: a _________________ and a

____________________________

i. Hypotenuse – ___________________________________

ii. Opposite Leg – __________________________________

iii. Adjacent Leg – _________________________________

IV. Review: Pythagorean Theorem

a. Pythagorean Theorem is used to find missing sides in a triangle.

b. “a” and “b” represent the _________________________________

c. “c” represents the ___________________________

d. Examples: Find the missing sides using Pythagorean Theorem

1. 2.

3. 4.


4


5

2 ft

6 ft

LK

J

x

103

x + 9

x + 7

x

Practice Problems

Use the Pythagorean Theorem to solve for the missing sides in the following triangles.

7. JK = ________ 8. x = ________ 9. x = ________

Inequalities in Triangles

10. Tell whether the given lengths may be the measures of the sides of a triangle:

a. 6 in., 4 in., 10 in. b. 6 in., 4 in., 12 in. c. 6 in., 4 in., 8 in.

11. Tell which of the following number triples may not be used as the lengths of the sides of a triangle:

a. (7, 8, 9) b. (3, 5, 8) c. (8, 5, 2) d. (3, 10, 6) e. (6, 9, 10)

12. The lengths of two sides of a triangle are 3 in. and 6 in. The length of the third side may be:

a. 3 in. b. 6 in. c. 9 in. d. 12 in.

13. Which of the following number triples cannot represent the length units of the sides of a triangle?

a. (2, 3, 4) b. (3, 1, 1) c. (3, 4, 5) d. (3, 4, 4)

14. In ΔABC, AB = 8, BC = 10, and CA = 14. Name the largest angle of triangle ABC.

15. In ΔABC, angle C contains 60° and AB is greater than AC. Angle B contains

a. 60° b. less than 60° c. more than 60°

16. In ΔABC, CA > CB and 𝑚∠B = 35. Angle C is

a. an acute angle b. a right angle c. an obtuse angle

Leg #1 Leg #2 Hypotenuse

1. 6 8

2. 5 12

3. 10 11

4. 14 10

5. 15 25

6. 3

10

1

2


6

13 12

12

3

21

x - 1

x + 1

x

3

2

1

60

(k + 2)kA B

C 10

9

8

D E

C

B

A

17. In ΔABC, 𝑚∠C = 90 and 𝑚∠B = 35. Name the shortest side of this triangle.

18. In ΔABC, 𝑚∠A = 74 and 𝑚∠B = 58. Which is the longest side of the triangle?

19. If in ΔRST, 𝑚∠R = 71 and 𝑚∠S = 37, then

a. ST > RS b. RS > RT c. RS = RT d. RT > ST

20. In ΔRST, ∠R is obtuse and 𝑚∠S = 50. Name the shortest side of the triangle.

In 21-24, fill in the blank with the word sometimes, always, or never, to make the statement true.

21. In ΔABC, ∠A contains more than 60°. Side 𝐵𝐶̅̅ ̅̅ is ______________________ the longest side of the triangle.

22. If one angle of a scalene triangle contains 60°, the side opposite this angle is __________________ the longest side

of the triangle.

23. If one of the congruent sides of an isosceles triangle is longer than the base, then the measure of the angle opposite

the base is _______________________ greater than 60°.

24. In ΔABC, if AB is greater than AC, then 𝑚∠C is __________________________ greater than 𝑚∠B.

The side lengths of two sides of a triangle are given. If x is the third side, complete the inequality to show the possible

values of x.

25. 6, 9 26. 15, 13 27. 100, 100 28. k, k + 5

_____ < x < _____ _____ < x < _____ _____ < x < _____ _____ < x < _____

The diagrams are not drawn to scale. If each diagram were drawn to scale, which numbered angle would be the largest?

29. 30.

The diagrams are not drawn to scale. If each diagram were drawn to scale, which segment would be the longest?

31. 32.


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Day 2: Special Right Triangles

Warm-Up:


8

Discovering Special Right Triangle Relationships

1. What are the measures of <B and <C? Explain how you arrived at your answer.

2. Classify the triangle by its angles and sides.

3. Write the Pythagorean Theorem. Write an equation in terms of x and h by substituting

the side lengths in the formula.

4. Solve for h. Your answer must be in simplified radical form.

5. Redraw the triangle above, substituting your answer from step 4 for h on the diagram.

6. Summarize your findings in a complete sentence.


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7. Draw a ⧍ DEF similar to ⧍ ABC where <D is the right angle, DE=5 and DF=5.

8. What is the measure of <E? Explain your answer.

9. What is the measure of <F? Explain your answer.

10. What is the length of EF? Explain how you found your answer. Does this answer coincide

with your findings in #6?

11. In every 45-45-90 triangle, the relationships of the sides is _________________.

For each problem, find the lengths of the other two sides of the triangle. Please put the

answers in simplified radical form.


10


11

Deriving the second type of Special Right Triangle

1. ⧍ ABC is an equilateral triangle. What is the measure of:

<A= <B= <ACB=

2. 𝐶𝐷̅̅ ̅̅ is an altitude of ⧍ ABC. What does 𝐶𝐷̅̅ ̅̅ do to segment AB?

3. Based on your answer to problem #2, find the following lengths in terms of x and write

them on the diagram.

AB= AD= BD=

4. 𝐶𝐷̅̅ ̅̅ is an altitude of the equilateral ⧍ABC. What does 𝐶𝐷̅̅ ̅̅ do to <ACB?

5. Find the following angle measures: m<1= m<2=

6. Highlight ⧍DBC. It is a 30-60-90 triangle. Write the Pythagorean Theorem. Write an

equation, substituting the side lengths of 𝐶𝐷̅̅ ̅̅ , 𝐶𝐵̅̅ ̅̅ , and 𝐷𝐵̅̅ ̅̅ .

7. Solve the equation written in step 4 for h. Leave your answer in simplest radical form.


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8. Redraw the triangle above, substituting your answer from step 5 for h on the diagram.

9. Summarize your findings in a sentence.

10. Fill in the following sentences:

a. In every 30- 60 -90 triangle, the hypotenuse is ___________ compared to the

side opposite the 30º angle.

b. In every 30-60-90 triangle, the side opposite the 60º angle is ___________

compared to the side opposite the 30º angle.

For each problem, find the lengths of the other two sides of the triangle. All answers must be

in simplest radical form.


13


14

Day 3: Special Right Triangles (Continued)

Warm-Up:


15

Word Problems/Applications

Solve each of the following. Draw a picture when necessary.

Leave all answers in simplified radical form.

1. The bottom of a ladder must be placed 3 feet from a wall. The ladder is 12 feet long.

How far above the ground does the ladder touch the wall?

2. A soccer field is a rectangle 90 meters wide and 120 meters long. The coach asks

players to run from one corner to the corner diagonally across the field. How far do the

players run?

3. Ryan quit bowling and took up sailing. His sail for his sailboat is a 45-45-90 Right

Triangle. The base of the sail is 6 feet long. What would the height of the sail be?

What is the length of the hypotenuse?

4. How far from the base of the house do you need to place a 15’ ladder so that it exactly

reaches the top of a 12’ wall?

5. Joe saw a “Yield” sign and “borrowed it.” He wanted to hang it up in his room because it

looked cool and it was in the shape of an Equilateral Triangle. The length of one side is

34 inches. What is the height of the sign?

6. What is the length of the diagonal of a 10 cm by 15 cm rectangle?


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7. The diagonal of a rectangle is 25 in. The width is 15 in. What is the area of the

rectangle?

8. Tristan has a square backyard with an area of 225 ft2. He started to plant grass seed

but only did half his yard. (He wanted to play GTA5 Heists instead) What is the

perimeter of the grass section of the backyard?

9. Two sides of a right triangle are 8” and 12”.

a. Find the area of the triangle if 8 and 12 are legs.

b. Find the area of the triangle if 8 and 12 are a leg and a hypotenuse.

10. Lorena and Karla are creating an art project in the shape of a right triangle. They have

a 92 cm long piece of wood, which is to be used for the hypotenuse. The two legs of the

triangular support are of equal length. Approximately how many more centimeters of

wood do they need to complete the support?

11. The area of a square is 81 cm2. Find the perimeter of the square.


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12. An isosceles triangle has congruent sides of 20 cm. The base is 10 cm. What is the area

of the triangle?

13. A baseball diamond is a square that is 90’ on each side. If a player throws the ball from

2nd base to home, how far will the ball travel?

14. Mr. Boyette has a tree farm. Half the farm is trees that he uses to make pencils, the

other half are maple trees that he uses to make “Ette’s Sweet Love Maple Syrup.” The

farm is a square divided into 2 sections along a 400 foot diagonal. What is the area of

the Maple Tree Farm section?

15. Jill’s front door is 42” wide and 84” tall. She purchased a circular table that is 96

inches in diameter. Will the table fit through the front door?


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Day 4: Quiz! Warm-Up:


19

Day 5: Trigonometric Ratios Warm-Up:


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21


22


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Day 6: Solve using Trigonometric Ratios Warm-Up:


24

Chant – “Picture Set-up Work Answer”

1. A ladder 14 feet long rests against the side of a building. The base of the ladder rests on

level ground 2 feet from the side of the building. What angle does the ladder form with the

ground?

2. A 24-foot ladder leaning against a building forms an 18˚ angle with the side of the building.

How far is the base of the ladder from the base of the building?

3. A road rises 10 feet for every 400 feet along the pavement (not the horizontal). What is the

measurement of the angle the road forms with the horizontal?

4. A 32-foot ladder leaning against a building touches the side of the building 26 feet above the

ground. What is the measurement of the angle formed by the ladder and the ground?

5. The directions for the use of a ladder recommend that for maximum safety, the ladder

should be placed against a wall at a 75˚ angle with the ground. If the ladder is 14 feet long,

how far from the wall should the base of the ladder be placed?


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Day 7: Solve using Trigonometric Ratios Warm-Up:


26

Day 8: Solving Right Triangles Warm-Up:


27

Example 1: Given the triangle below, which of the following methods could be used to solve for x, y, and/or z

Pythagorean Theorem Yes or No

If yes, which variable(s) can I solve for _________

30-60-90 Triangle Yes or No




Right Triangle Trigonometry Yes or No




















30°

y

x

12

z

y

z

x

16 20

57°

y x

12 z


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You try #1: Solve for x, y, and z using the method of your choice.

x = _______________

y = _______________

z = _______________

You try #2: Solve for x, y, and z using a different method, formula, or strategy than you did in You Try #1.

x = _______________

y = _______________

z = _______________

What impact did your choice of method have on your final answer and the overall difficulty of the problem?

__________________________________________________________________________________________________

__________________________________________________________________________________________________


x = _______________

y = _______________

z = _______________


x = _______________

y = _______________

z = _______________

30°

y

x

12

z

30°

y

x

12

z

57°

y x

12 z

y

z

x

16 20


29

Practice Problems

1. The slide at the playground has a height of 6 feet. The base of the slide measured on the ground

is 8 feet. What is the length of the sliding board? What is the angle between the ground and the

slide?

2. The bottom of a 13-foot straight ladder is set into the ground 5 feet away from a wall. When the

top of the ladder is leaned against the wall, what is the distance above the ground it will reach?

What is the angle between the ladder and the wall?

3. In shop class, you make a table. The sides of the table measure 36" and 18". If the diagonal of

the table measures 43", is the table “square”? (In construction, the term "square” just means the

table has right angles at the corners.)

4. In the Old West, settlers made tents out of a piece of cloth thrown over a clothesline and then

secured to the ground with stakes forming an isosceles triangle. How long would the cloth have

to be so that the opening of the tent was 6 feet high and 8 feet wide? What is the measure of the

angle formed by the two sides of cloth where the cloth is thrown over the clothesline?

5. A baseball “diamond” is actually a square with sides of 90 feet. If a runner tries to steal second

base, how far must the catcher, at home plate, throw to get the runner “out”? Given this

information, explain why runners more often try to steal second base than third.

6. Your family wants to purchase a new laptop with a 17” widescreen. Since the 17 inches represents

the diagonal measurement of the screen (upper corner to lower corner), you want to find out the

actual dimensions of the laptop. When you measured the laptop at the store, the height was 10

inches, but you don’t remember the width. Calculate and describe how you could figure out the

width of the laptop to the nearest tenth inch.

7. During a football play, DeSean Jackson runs a straight route 40 yards up the sideline before

turning around and catching a pass thrown by Michael Vick. On the opposing team, a defender

who started 20 yards across the field from Jackson saw the play setup and ran a slant towards

Jackson. What was the distance the defender had to run to get to the spot where Jackson caught

the ball?

8. In construction, floor space must be planned for staircases. If the vertical distance between the

first and second floors is 3.6 meters, and a contractor is using the standard step pattern of 28 cm

wide for 18 cm high, then how many steps are needed to get from the first to the second floor

and how much linear distance (ie “width” or “base”) will be needed for the staircase? What is the

length of the railing that would be attached to these stairs? At what angle should the railing be

hung?


30

Day 9: Angles of Elevation & Depression

Warm-Up: Draw a picture to model each question. DO NOT SOLVE.

While watching the launch of a NASA space shuttle, a 6’ tall student stands at a point away from the base of the launch (he doesn’t know how far away due

to security). He knows the space shuttle is 184 feet standing on the launch pad and measures the angle of elevation to the top of the shuttle to be 22°.

How far away from the space shuttle launch pad does security make observers stand?

A cable car at a ski resort carries skiers to a height of 3 kilometers. If the cable of the car is 8 kilometers in length, what is the angle of depression of the cable from the top of

the mountain to the ski resort at the bottom?

You are working as an engineer at a water park. You are building a waterslide, and you have a few restrictions. The ladder up the slide

has to be between 20 and 35 feet tall. The slide has to make a 50°angle of elevation with the ground. It costs $100 per foot of slide

when the slide is under 30 feet and $70 per foot of slide when the slide is 30 feet or longer, and $25 per foot of ladder. Is it cheaper to

build the longer slide or the shorter slide? If you owned the park, which would you choose to build?


31

Angles of Elevation and Depression

The angle of elevation is the angle formed by a _________________ and the

line of sight looking up.

The angle of depression is the angle formed by a _______________ and the line

of sight looking down.

Notice … the angle of elevation and the angle of depression are

_____________________________ when in the same picture!


32

Angles of Elevation & Depression

Find all values to the nearest tenth.

1. A man flies a kite with a 100 foot string. The angle of elevation of the

string is 52 o . How high off the ground is the kite?

2. From the top of a vertical cliff 40 m high, the angle of depression of an object that is level with the base of

the cliff is 34º. How far is the object from the base of the cliff?

3. An airplane takes off 200 yards in front of a 60 foot building. At what angle of elevation must the

plane take off in order to avoid crashing into the building? Assume that the airplane flies in a

straight line and the angle of elevation remains constant until the airplane

flies over the building.

4. A 14 foot ladder is used to scale a 13 foot wall. At what angle of elevation must the ladder be situated in

order to reach the top of the wall?

5. A person stands at the window of a building so that his eyes are 12.6 m above the level ground. An object is

on the ground 58.5 m away from the building on a line directly beneath the person. Compute the angle of

depression of the person’s line of sight to the object on the ground.

6. A ramp is needed to allow vehicles to climb a 2 foot wall. The angle of elevation in order for the vehicles to

safely go up must be 30 o or less, and the longest ramp available is 5 feet

long. Can this ramp be used safely?


33

Draw a picture, write a trig ratio equation, rewrite the equation so that it is calculator ready and then solve each problem. Round measures of segments to the nearest tenth and measures of angles to the nearest

degree.

________1. A 20-foot ladder leans against a wall so that the base of the ladder is 8 feet from the base of the building. What is the ladder’s angle of elevation?

________2. A 50-meter vertical tower is braced with a cable secured at the top of the tower and tied 30 meters from the base. What is the angle of depression from the top of the tower to the point on the ground where the cable is tied?

________3. At a point on the ground 50 feet from the foot of a tree, the angle of elevation to the top of the tree is 53. Find the height of the tree.

________4. From the top of a lighthouse 210 feet high, the angle of depression of a boat is 27. Find the distance from the boat to the foot of the lighthouse. The lighthouse was built at sea level.

________5. Richard is flying a kite. The kite string has an angle of elevation of 57. If Richard is standing 100 feet from the point on the ground directly below the kite, find the length of the kite string.

________6. An airplane rises vertically 1000 feet over a horizontal distance of 5280 feet. What is the angle of elevation of the airplane’s path?

________7. A person at one end of a 230-foot bridge spots the river’s edge directly below the opposite end of the bridge and finds the angle of depression to be 57. How far below the bridge is the river?

________8. The angle of elevation from a car to a tower is 32. The tower is 150 ft. tall. How far is the car from the tower?

230


34

________9. A radio tower 200 ft. high casts a shadow 75 ft. long. What is the angle of elevation of the sun?

________10. An escalator from the ground floor to the second floor of a department store is 110 ft long and rises 32 ft. vertically. What is the escalator’s angle of elevation?

________11. A rescue team 1000 ft. away from the base of a vertical cliff measures the angle of elevation to the top of the cliff to be 70. A climber is stranded on a ledge. The angle of elevation from the rescue team to the ledge is 55. How far is the stranded climber from the top of the cliff? (Hint: Find y and w using trig ratios. Then subtract w from y to find x)

________12. A ladder on a fire truck has its base 8 ft. above the ground. The maximum length of the ladder is 100 ft. If the ladder’s greatest angle of elevation possible is 70, what is the highest above the ground that it can reach?

________13. A person in an apartment building sights the top and bottom of an office building 500 ft. away. The angle of elevation for the top of the office building is 23 and the angle of depression for the base of the building is 50. How tall is the office building?

________14. Electronic instruments on a treasure-hunting ship detect a large object on the sea floor. The angle of depression is 29, and the instruments indicate that the direct-line distance between the ship and the object is about 1400 ft. About how far below the surface of the water is the object, and how far must the ship travel to be directly over it?

110 32

1000

x

w y

8

100

500

1400


35

Day 10: Angles of Elevation & Depression

Warm-Up:

1. An observer in a 55 feet tall lighthouse spots a ship in distress at an angle of

depression of 9.5 degrees. How far is the ship from the shore?

2. You are standing on a platform ready to swing from a rope attached to a tree in

order to swing into the Little River. Your friend is waiting in the river below. She

is across from your position 25 feet from the edge of the river. You spot her at an

angle of depression of 56 degrees. Approximately how far will you drop before

you splash into the river below?


36

Day 11: Clinometer Activity

Warm-Up:

Note that angles of depression and elevation are

measured from a horizontal line. Also, because they

are alternate interior angles of parallel lines they are

congruent.

1. At a certain time of day the angle of elevation of the sun is 44°. Find the length of the

shadow cast by a building 30 meters high.

2. The top of a lighthouse is 120 meters above sea level. The angle of depression from the top

of the lighthouse to the ship is 23°. How far is the ship from the foot of the lighthouse?

3. A lighthouse is 100 feet tall. The angle of depression from the top of the lighthouse to one

boat is 24°. The angle of depression to another boat is 31°. How far apart are the boats?

4. To the nearest tenth, find the measure of the acute angle that the line forms with a

horizontal line.

y = 2

3x + 1

Angle of depression

Angle of elevation

y

x


37

Day 12: Unit Review

Warm-Up:

For the next five minutes, write down EVERYTHING you know about

the following topics:

1. Pythagorean Theorem

2. 45-45-90 Triangles

3. 30-60-90 Triangles

4. Right Triangle Trigonometry

honors math 2 unit 6 class packet sanderson high …...1 day 1: pythagorean theorem and converse...

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