In Unit 1 you worked with some similar figures and observed that corresponding angles are equal and that the corresponding sides have a constant ratio or scale factor.
In this activity, you will focus on similar triangles.
Complete the following activity on identifying similar triangles.
Open your Word Journal.
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In this activity, the Mathematical Processes Connecting and Reasoning and Proving are both addressed.
Open your document U2 Mathematical Processes and read the descriptions for the processes.
As you complete the activity, notice when you are inferring, justifying and concluding (as part of the work or as part of your thinking) and insert your record below the description of the process.
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This activity is an introduction to the study of trigonometry trigonometry (definition:measurement of angles and sides in triangles; trigon - meaning triangles and metry - referring to measurement), a branch of geometry. Trigonometry is used in many fields of study including music, gaming, construction and astronomy. To become familiar with a few of the interesting applications of trigonometry, watch the following video.
You looked at different sets of 2 similar triangles and, although each set was different, they had similar properties. You verified that:
The properties apply whether you are looking at an acute, obtuse or right angle triangle.
In this activity, you are going to focus on right triangles. A right triangle has special properties. Also, you are going to learn about the ratios between two sides within one triangle instead of between two triangles.
There are different ways to identify the sides in a triangle
You may be used to identifying the sides in a triangle by using the line segments identified by the vertices (definition:Vertices are the points where two line segments meet. Note that the vertex of a triangle is not the same as the vertex of a parabola, although we use the same term for both.)
Another way that we can refer to a side of a right triangle is by relating the side to the angle across from it.
One name that you may already be familiar with is the hypotenuse(definition:The side of a triangle that is across from the angle of 90 (degree symbol) is called the hypotenuse of the triangle.). It is always the longest side in the triangle.
The names for the other two sides in a right triangle are opposite and adjacent.
The labels, opposite and adjacent, are applied with reference to one of the acute angles in the triangle. Unlike the hypotenuse, the labels can change.
The following interactive will help you understand the naming of the sides in a right triangle:
By investigating triangles you will see that the constant ratio is connected to a particular angle value. The ratios exist for all degree values.
Each name is attached to one ratio.
Complete the bottom three rows of the table with the full name, the abbreviation that is used and the definition.
To complete the table in U2A3 Record of Ratios, you will find values using a scientific or graphing calculator or a math app on your computer, phone, or tablet.
Verify that it is working in degrees (not radians).
Please note that not all calculators or software operate the same! Ask your teacher for help with checking the degree/radian setting and how to use the sin, cos and tan buttons that you will use below.
Find the buttons sin, cos and tan on your device.
Press sin then 20° (or 20 then sin for some calculators) to get the value of sin 20°.
This value is already recorded in the Record of Ratios table. Next, use button cos to find cos 20° and use tan to find tan 20°.
Work down and across the table using the angle measure shown for each row and using the same button in each column sin, cos and tan.
The values from the calculator should be the same as the values in your tables.
These new ratios are called trigonometric ratios. You will be learning how to use these three primary trigonometric ratios in the next and future activities.
You may hear reference to the mnemonic (definition:a pattern of letters that helps in remembering something.) SOH CAH TOA.The first letter represents the name of the ratio and the second and third letters represent the sides of the triangle in the ratio.
You can decide if you would like to use it.
For further study of the three primary trig ratios, view the following:
Basic Trigonometry (Khan Academy)Download and complete the page, U2A3 What’s My Ratio.
When you have completed the page, compare your work to the answer sheet to determine whether you have met the Success Criterion for this activity, U2A3 What’s My Ratio answers.
In this activity, the Mathematical Processes Connecting and Reasoning and Proving were the focus.
Open your document U2 Mathematical Processes and complete your record of when you used the process of representing.
Save to your Portfolio.
Open your document Unit 2 Reflections.
Reflect on the new ideas, vocabulary, graphs etc that you learned about in this activity.
For this reflection, use the stems:
to write your reflections for this activity.
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Congratulations, you have completed Unit 2, Activity 3. You may move on to Unit 2, Activity 4.