 Use the interactive below to create a sequence of similar triangles. Use the results from the interactive to complete the table that follows and answer the questions.

Copy and complete the table comparing the ratios of the lengths of the given sides of the triangles above into your notes or onto your own sheet of paper.

 Original Segment ΔDEF ΔGHI ΔJKL AB DE over AB DE AB = GH over AB GH AB = JK over AB JK AB = BC EF over BC EF BC = HI over BC HI BC = KL over BC KL BC =

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 Original Segment ΔDEF ΔGHI ΔJKL AB DE over AB DE AB = 6 over 3 6 3 = 2 GH over AB GH AB = 9 over 3 9 3 = 3 JK over AB JK AB = 12 over 3 12 3 = 4 BC EF over BC EF BC = 4 over 2 4 2 = 2 HI over BC HI BC = 6 over 2 6 2 = 3 KL over BC KL BC = 8 over 2 8 2 = 4 ### Pause and Reflect

Inductive reasoning is a thought process by which rules are generated from observations of patterns. How did you use inductive reasoning to generate your expression to determine the lengths of the legs of Triangle z in the sequence of similar triangles above?

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You used inductive reasoning when you used a sequence of triangles to generate a table of data, and then used your table of data to generate an expression that served as a rule for determining the lengths of the legs for any triangle. ### Practice

Use the information below about a sequence of similar triangles to answer the questions that follow. You may want to sketch the triangles to help you answer the questions.

Triangle 1: 3 cm, 3 cm, 4 cm
Triangle 2: 6 cm, 6 cm, 8 cm
Triangle 3: 12 cm, 12 cm, 16 cm
Triangle 4: 24 cm, 24 cm, 32 cm

1. If Triangle 1 is the original triangle, what scale factor is used to generate Triangle 2, Triangle 3, and Triangle 4 from Triangle 1?

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What factor do you need to multiply the lengths the sides of Triangle 1 by in order to generate the lengths of the sides of the other triangles? Interactive popup. Assistance may be required.

Triangle 2: scale factor = 2
Triangle 3: scale factor = 4
Triangle 4: scale factor = 8 2. What pattern do you observe in the scale factors?

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How are the numbers 2, 4, and 8 related? Interactive popup. Assistance may be required.

The scale factor is a sequence of powers of 2. 3. How do the scale factors relate to the triangle number?

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Write the scale factors as powers of 2 using exponents. Interactive popup. Assistance may be required.

Triangle 2: scale factor = 2 = 21
Triangle 3: scale factor = 4 = 22
Triangle 4: scale factor = 8 = 23
The power of 2 used for the scale factor is 1 less than the triangle number. 4. Use the pattern to predict the scale factor that could be used to generate the side lengths of Triangles 5 and 6.

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What are the next numbers in the sequence of scale factors? Interactive popup. Assistance may be required.

Triangle 5: scale factor = 24 = 16
Triangle 6: scale factor = 25 = 32 5. Write an expression that could be used to generate the side lengths of Triangle n from the side lengths of Triangle 1 in this sequence of similar triangles.

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If the triangle number is n, what power of 2 do you need to use for the scale factor? Interactive popup. Assistance may be required.  