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Math 2 Module 6: Similarity & Right Triangle Trigonomtry

6.1 Ready, Set, Go!

Questions 1 - 6

There are no missing lengths here, but think about how you'd find out the scale factors for the triangle's sides: if one side was 1.5 units in the pre-image and 3 units in the image, what's the scale factor? Here's a video to help: https://www.youtube.com/watch?v=fA2EZoYgMgA

Give the factor by which each pre-image was multiplied to create the image. Use the scale factor to fill in any missing lengths.

Questions 7 - 12

Determine center of dilation and tool being used. These questions will make you think creatively. Here is a video explaining dilation in more mathematical terms that may help: 

https://www.youtube.com/watch?v=BCllaARDOWI
 

Questions 13 - 18

ITest the points in the table to determine if the slope is constant. What kind of graph from the three options has a constant slope? What kinds of graphs from the three options do not have a constant slope. If you're struggling, here's a video on function types that can help: https://www.youtube.com/watch?v=Air3gz2KabA

Determine whether the given representation is representative of a linear, exponential, or quadratic function, classify it as such and justify your reasoning.

6.2 Ready, Set, Go!

Questions 1 - 6

This video reviews different types of angles at intersections of lines: https://www.youtube.com/watch?v=dA94zyaLuhk

Questions 7 - 10

This video shows you how to dilate (scale up or down) an image around a point: https://www.youtube.com/watch?v=4Yap2t_v034

Questions 11 - 14

This video shows you how to find the scale factor and center of dilation of two similar shapes: https://www.youtube.com/watch?v=M5dsbRwVdLY
 

Questions 15 - 18

Determine the transformations (rotation, dilation, translation, etc) that occurred. This video walks through shape transformations: https://www.youtube.com/watch?v=XiAoUDfrar0

6.3 Ready, Set, Go!

Questions 1 - 9

Solve the proportion for c in the denominator. This video shows you how to solve similar proportions: https://www.youtube.com/watch?v=he8Gl03cZHg

For proportions with a binomial, check out this video: https://www.youtube.com/watch?v=--tlBznmr9o

Question 10

Triangle similarity. This (really old) video has some helpful information within the first minute that will help you think about this: https://www.youtube.com/watch?v=1UuiF3mskQA

Show that not all right triangles are similar. Think of an example.

Question 11

Regular polygon similarity. What's a regular polygon specifically? Being clear with the definition can help better tackle the problem. See here: https://www.youtube.com/watch?v=jS1W44Pl0ek

Here's a cool video on showing a polygon as sides increase: https://www.youtube.com/watch?v=x2m6J6MLixI

Questions 12 - 13

While orientation doesn't matter, think about if the scale factor to each side has been held constant. To help you think more about each set, watch this video: https://www.youtube.com/watch?v=7aDPe2wvQAI

Questions 14 - 17

Determine transformations (rotation, dilation, translation, etc). Think about how the overall shape moved and how individuals points transformed. How would you recreate this change (to answer the question)? If you need additional help, this video may be an aid: https://www.youtube.com/watch?v=XiAoUDfrar0

Questions 18 - 21

Write equivalent ratios. For this problem, you need to know how to find the scale factor and think of equivalent ratios. For some more guidance on equivalent ratios, see here: https://www.youtube.com/watch?v=WuiTHW2hVEc

6.4 Ready, Set, Go!

Questions 1 - 4

Using the Pythagorean theorem, we can find one side of a right triangle if we know how long the other two sides are. If you need a refresher, check out this video: https://www.youtube.com/watch?v=AA6RfgP-AHU

Questions 5 - 6

Write and solve a proportion. You have all the information you need. Set up proportions per triangle where numerators and denominators correspond to the same sides of different triangles. If you need some guidance, watch this: https://www.youtube.com/watch?v=he8Gl03cZHg

Questions 7 - 8

Write equal ratios. For this problem, there are no numbers to think about. Instead, think about how the relationships per letter vary depending on the angles they're next to where you can compare them. For some more guidance on equivalent ratios, see here: https://www.youtube.com/watch?v=WuiTHW2hVEc

Questions 9 - 10

Write and solve a proportion. You have all the information you need. Choose your numerator and denominator, and make sure that in both of your equations they correspond to the same lengths per shape. If you need some guidance, watch this: https://www.youtube.com/watch?v=he8Gl03cZHg

Questions 11 - 13

Write a similarity statement. Do you think there's a scale factor? Do you see the angles maintained between triangles? Think about these questions as you make your similarity statement. Here's a more detailed video to guide you: https://www.youtube.com/watch?v=mHjGS3IOVZk

Questions 15 - 18

There are a couple of ways to do this question. I proved that the slope is constant using the similar triangles method, like this video: https://www.youtube.com/watch?v=24WMbh1BBKc

6.5 Ready, Set, Go!

Questions 1 - 6

These are all right triangles, so you can apply the Pythagorean theorem to solve for the missing side! For a refresher, watch this video: https://www.youtube.com/watch?v=AA6RfgP-AHU

Question 7

Proportional relationships to find desired values. It'll help to think about what a similar triangle will look like to answer most of these questions. So, think about scale factors and gut intuition. Here's a refresher video on similar triangles and slope, which may help in graphs: https://www.youtube.com/watch?v=24WMbh1BBKc

Questions 8 - 9
Understanding angle relationships here is key to find the missing values. Then, use your previous knowledge on similar shapes and triangles to find the missing lengths. If you need a refresher on angle hunting, watch this: https://www.youtube.com/watch?v=zn_svvBOci4

In each of the diagrams use the given information provided to find the missing lengths and angle measurements

Questions 10 - 18

Though not all of them do, most of these problems have multiple x's you have to deal with to solve for the final x. You're going to need to make solving for one x a priority, and the rest is algebra you know. For some more basic examples, watch this: https://www.youtube.com/watch?v=P_aQPxr2G_4

6.6 Ready, Set, Go!

Questions 1 - 6

To calculate the mean, just add the numbers and divide by how many numbers there are. How does the number you find compare to the set of numbers in total? Here's a refresher on finding means: https://www.youtube.com/watch?v=DZYhvMCFE5Q

Questions 7 - 12

Calculate halfway point. The halfway point is the same as the average of two numbers. Here's a quick tutorial if you need it: https://www.youtube.com/watch?v=n83F6wmmWTI

Find the value that is exactly halfway between the two given values. Explain how you find this value.

Questions 13 - 18

You have to find the midpoint of the line given the endpoints of the line. This video shows you how: https://www.youtube.com/watch?v=Ez_-RwV9WVo

Find the coordinates of the midpoint of each line segment below. If multiple line segments are given then give the midpoints of all segments.

Questions 19 - 22

Proportional relationships to find desired values. It'll help to think about what a similar triangle will look like to answer most of these questions. So, think about scale factors and gut intuition. Here's a refresher video on similar triangles: https://www.youtube.com/watch?v=BI-rtfZVXy0

Question 23

To think about this, use some of the triangles above to test out your ideas. What are you noticing?

Question 24

Test out some ideas on the graphs above. Ask yourself, are there similar triangles that aren't right triangles? What might the answer to that hint?

Questions 25 - 26

Write and solve a proportion. Think about which sides you can compare to set up your proportion and find the variable. Here is an example of someone doing something similar: https://www.youtube.com/watch?v=mjqQAIp9-Fo

6.7 Ready, Set, Go!

Question 1

Identify similar and congruent triangles. Pay attention to what the symbols tell you about angle and lengths. Here's a video to refresh the differences between similar and congruent triangles: https://www.youtube.com/watch?v=0mKmYxuMirE

Determine which of the triangles below are similar and which are congruent. Justify your conclusions. Give your reasoning for the triangles you pick to be similar and congruent.

Questions 2 - 7

Missing value in the proportion. Using the big triangle, work out all these relationships, which will be very helpful in the following problems. For a refresher on proportions in similar triangles, watch this: https://www.youtube.com/watch?v=eybwWz3Ntns

Use the given right triangles with altitudes drawn to the hypotenuse to correctly complete the proportions.

Questions 8 - 9

Pythagoras missing sides. Use the relationships from your previous answers to think about how to solve the proportion for the missing sides. ​

Questions 10 - 11

Find missing values in similar triangles. Think back to dilations and scale factor to help with your proportion. If you need a refresher, watch this: https://www.youtube.com/watch?v=rgup4S56RYo

Questions 12 - 15

Missing values in a table. To find the missing values in the table, you need to understand arithmetic and geometric sequences to find the pattern and apply it to the blank space. Watch this for help: https://www.youtube.com/watch?v=fHX3IxwJ61M

6.8 Ready, Set, Go!

Questions 1 - 6

Because these are all right triangles, you can use the Pythagorean theorem to find the missing side. And, because the angles in a triangle always sum up to 180 degrees, you can solve for the missing angle!

 

Watch this video for a refresher on Pythagorean problems: https://www.youtube.com/watch?v=YlDfGFBNPmA

 

To find missing angles using the sum of interior angles, check out this video: https://www.youtube.com/watch?v=Dnu-gPjaIdY

 

Questions 7 - 10

Trigonometric ratios. Use your pre-existing knowledge on the Pythagorean theorem to solve for the missing side lengths. Then, use trigonometric ratios to find the sin/cos/tan of the given angles. Here's a thorough video with an example: https://www.youtube.com/watch?v=Jsiy4TxgIME

 

Write desired trig ratios and find missing sides of the triangle if any are present.

Questions 11 - 19

Factor the quadratic equation first. Then, find which inputs make y = 0 to find your x-intercepts, and plug in x = 0 to find the y-intercept. Here's a helpful video with a few factoring examples: https://www.youtube.com/watch?v=YtN9_tCaRQc

 

Show the factored form of the quadratic and its intercepts.

6.9 Ready, Set, Go!

Question 1

Think about these formulas and feel free to draw helpful diagrams. As you write them, notice the relationship between them, like between the area of a rectangle and a rectangular prism's volume. Here's a refresher video on some volumes: https://www.youtube.com/watch?v=3wuJJqlr6m0

 

Question 2

You have all the values you would need to solve this, and you wrote the equation you need in the last problem. If you still need help, watch this: https://www.youtube.com/watch?v=O-cawByg2aA

 

Question 3

You have all the values you would need to solve this, and you wrote the equation you need in the first problem. If you still need help, watch this: https://www.youtube.com/watch?v=9HPWB8UX8GQ

 

Question 4

You have all the values you would need to solve this, and you wrote the equation you need in the first problem. If you still need help, watch this: https://www.youtube.com/watch?v=2rgSt84wm_Y

 

Questions 5 - 8

Though it looks like you don't have much, you have all you need. Use the first equation to do an initial sketch of the triangle, find the length of the missing side, and write down the rest of your trig ratios. Here's an example: https://www.youtube.com/watch?v=1mvU2ym9lDY

Question 9

Before getting tripped up in the equation, think about what it symbolizes. Look at your answers above and do some experiments to find the answer. It might also be helpful to write cosine as A/H and sin as O/H and solve to see if you get the same answer.

 

Question 10

It may help to have an example triangle written out. Think about what the equation is saying: is B equal to 90 - A? If so, then the whole equation could be correct. How will you find out?

 

Question 11

In this problem, it will be helpful to break each ratio down into its parts (ie. sin = opp/hyp) to see if you can cancel anything out and make tan A * cos A equal to sin A.

 

Questions 12 - 14

There are many first inputs for rise over run, but they should all lead to the same, simplified answer. Look at this video for an example: https://www.youtube.com/watch?v=2kMUk_XRvRQ

 

Questions 15 - 17

Use Pythagorean theorem knowledge to find the missing side, then following a rise/run model, find the slope. Remember, check your gut when you get your answer, especially on if it should be positive or negative. (Also remember the formula for slope y-y/x-x) Watch this for some guidance: https://www.youtube.com/watch?v=svbDTxFHHf8

6.10 Ready, Set, Go!

Questions 1 - 3

These problems are telling you essential information about a triangle that exists in real life. Weed out that key information to draw a triangle representing the situation, and use previous knowledge to answer the specific question. Here's a helpful video: https://www.youtube.com/watch?v=svbDTxFHHf8

 

Questions 4 - 7

Because these are all right triangles, you can use the Pythagorean theorem to find the missing side. And, because the angles in a triangle always sum up to 180 degrees, you can solve for the missing angle!

 

Watch this video for a refresher on Pythagorean problems: https://www.youtube.com/watch?v=YlDfGFBNPmA

 

To find missing angles using the sum of interior angles, check out this video: https://www.youtube.com/watch?v=Dnu-gPjaIdY

Question 11

You are given the first term and common ratio, r, for an geometric sequence g(r). You need to find the first 5 terms of the sequence with that information. This video shows you how to do a similar question - notice that you already have the common difference, so you don't need to calculate it: https://www.youtube.com/watch?v=3uiTQ6JSMC0

Questions 12 - 13

To find the explicit formula for a geometric sequence, check out this video: https://www.youtube.com/watch?v=faBM7b8KKjM

Question 14

Plug the value x = 100 into the explicit expressions you found in questions 11., 12., and 13. and see which value is largest!

6.11 Ready, Set, Go!

Questions 1 - 6

Write and solve a proportion. Think about similar shapes to get in the right mindset. You have all the numbers you need but don't get tripped up on side lengths by forgetting to subtract or add something that should or shouldn't be included in one shape's equation. If you need a refresher, watch this: https://www.youtube.com/watch?v=rgup4S56RYo

 

Use trigonometric ratio to sketch the right triangle and solve the missing side.

 

Questions 7 - 12

Because these are all right triangles, you can use the Pythagorean theorem to find the missing side. And, because the angles in a triangle always sum up to 180 degrees, you can solve for the missing angle!

 

Watch this video for a refresher on Pythagorean problems: https://www.youtube.com/watch?v=YlDfGFBNPmA

 

To find missing angles using the sum of interior angles, check out this video: https://www.youtube.com/watch?v=Dnu-gPjaIdY

 

Questions 13 - 18

Use the first equation to do an initial sketch of the triangle, find the length of the missing side, and use the 180 angle sum knowledge or trig ratios to find the missing angle. This video shows a different but similar process that shows helpful steps: https://www.youtube.com/watch?v=1mvU2ym9lDY

Questions 19 - 30

Evaluate trigonometric expressions. These questions challenge you to creatively consider everything you have learned. Think thoroughly about sin, cos, and tan, including how their ratios break down. Some of these questions need you to think about the Pythagorean theorem. Test out your ideas by writing equations and checking your gut.

 

This video goes through more trig ratios that may provide insight: https://www.youtube.com/watch?v=G-T_6hCdMQc

Question 31

Sketch the situation to find the triangle that you may feel more familiar with. Finding the missing value will build on skills you've practiced in this module. This video covers a similar problem: https://www.youtube.com/watch?v=3H28-wzsF3s

 

Question 32

Sketch the situation to find the triangle that you may feel more familiar with. Finding the missing value will build on skills you've practiced in this module. This video covers a similar problem: https://www.youtube.com/watch?v=ILQ9TC-0_DA

 

Question 32

Sketch the situation to find the triangle that you may feel more familiar with. Finding the missing value will build on skills you've practiced in this module. This video covers a similar problem: https://www.youtube.com/watch?v=y0b90Yz-LPM

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