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Exploring Similar Right Triangles: Activities and Proportions with Dr. Jennifer L. Brown

In this workshop led by Dr. Jennifer L. Brown, participants engage in hands-on activities to explore similar right triangles. Using simple materials like paper, rulers, and colored pencils, attendees draw, cut, and arrange triangles to identify relationships between angles and sides. The workshop emphasizes the concept of similarity, demonstrating how triangles can be proven similar through congruent angles and proportional sides. Participants will also tackle problems involving measurements and proportions, fostering a deeper understanding of geometric relationships.

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Exploring Similar Right Triangles: Activities and Proportions with Dr. Jennifer L. Brown

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  1. Similar Right Triangles Dr. Jennifer L. Brown, 2013, Columbus State University, CRMC Summer Workshop (MCC9‐12.G.SRT.4; MCC9‐12.G.SRT.5)

  2. Part 1

  3. Materials Needed • rectangular piece of paper • ruler • scissors • colored pencils

  4. Step 1: Draw a diagonal.

  5. Step 2: Draw an altitude.

  6. Step 3: Label the triangles. 1 9 2 8 5 7 3 6 4

  7. Step 4: Cut out the triangles. 1 9 2 8 5 7 3 6 4

  8. Step 5: Arrange the triangles using 1, 4, & 7. 1 1 9 2 2 8 5 7 3 3 6 4

  9. Questions How are the two smaller triangles related to the large triangle? Explain how you would show that the triangle that includes ∠8 is similar to the triangle that includes ∠5. Explain how you would show that the triangle that includes ∠5 is similar to the triangle that includes ∠2.

  10. Questions Write a proportion involving the two legs of the triangle that includes ∠2 and the triangle that includes ∠5. Measure the legs in centimeters (to the nearest tenth) and substitute the values. Cross-multiply. What did you notice? 21.6 cm 17.1 cm bottom leg 2 right leg 2 bottom leg 5 right leg 5 ____________ = ____________ 27.9 cm 22.0 cm

  11. 7. Could we find the lengths for the other sides without measuring? 1 27.9 cm 9 2 13.2 cm 8 35.3 cm 21.6 cm 21.6 cm 5 17.1 cm 7 22.0 cm 17.1 cm 3 6 4 27.9 cm

  12. B B B Z Z Z Y Y Y X X X A A A C C C 10 54o 91o 91o 10 12 6 6 91o 15 9 15 91o 9 54o 18 How Can Triangles Be Proven Similar? Similar () triangles have congruent () angles and proportional sides. Side – Side – Side (SSS) Angle – Angle (AA) Side – Angle – Side (SAS) A  X C  Z C  Z AB BC CA XY YZ ZX BC CA YZ ZX ___ ___ ___ ___ ___ = = = Dr. Jennifer L. Brown, 2013, Columbus State University, CRMC Summer Workshop

  13. B B B Z Z Z Y Y Y X X X A A A C C C 10 54o 91o 91o 10 12 6 6 91o 15 9 15 91o 9 54o 18 How Can Triangles Be Proven Similar? Similar () triangles have congruent () angles and proportional sides. Side – Side – Side (SSS) Angle – Angle (AA) Side – Angle – Side (SAS) Dr. Jennifer L. Brown, 2013, Columbus State University, CRMC Summer Workshop

  14. Part 2

  15. Directions • Draw a diagonal from the top left corner to the lower right corner. • Cut along the diagonal.

  16. Label ΔABC (as shown). • Label points D, E, F, & G. • Fold side BC up to meet point D. (Keep BC ⊥ to AB. ) • Label point E. • Draw segment DE. • Repeat step 5 but meet point F. • Label point G. • Draw segment FG. A E D G F B C

  17. A • Measure the length of AC, AB, and BC (in centimeters to the nearest tenth). • Measure the length of AG, AF, and FG. • Measure the length of AE, AD, and DE. E D G F B C

  18. Questions • What do you notice about the following lengths? • AGAF • AC AB Why?

  19. Questions • 2. What do you notice about the following lengths? • DEAE • BC AC Why?

  20. Questions • 3. What do you notice about the following lengths? • AGFG • AB DE Why?

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