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Chapter 5 Introduction to Trigonometry: 5.4 Modelling with Similar Triangles

Chapter 5 Introduction to Trigonometry: 5.4 Modelling with Similar Triangles. Humour Break. 5.4 Modelling with Similar Triangles. Goals for Today: Apply what we have learned about similar triangles to some word problems. 5.4 Modelling with Similar Triangles.

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Chapter 5 Introduction to Trigonometry: 5.4 Modelling with Similar Triangles

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  1. Chapter 5 Introduction to Trigonometry: 5.4 Modelling with Similar Triangles

  2. Humour Break

  3. 5.4 Modelling with Similar Triangles Goals for Today: • Apply what we have learned about similar triangles to some word problems

  4. 5.4 Modelling with Similar Triangles • Ex. 1 – Rachel is standing beside a lighthouse on a sunny day. Rachel is 1.6m tall and the sun casts a shadow 4.8m long (on the ground). At the same time, the sun casts a shadow 75m long (on the ground). How tall is the lighthouse?

  5. 5.4 Modelling with Similar Triangles

  6. 5.4 Modelling with Similar Triangles

  7. 5.4 Modelling with Similar Triangles • AB is ll (parallel) to DE (same sun’s rays) • AC is on the same line as DF (both flat on ground) • ∠BAC = ∠EDF (same ∠) • ∠ACB = ∠DFE (both 90°) • ∠ABC = ∠DEF (180° ∆ rule) • Therefore, ∆ABC ~ ∆DEF

  8. 5.4 Modelling with Similar Triangles Since ∆ABC ~ ∆DEF • Plug in the known values • Cross-multiply • Simplify & solve for x

  9. 5.4 Modelling with Similar Triangles • Ex. 2 – A 3.6 m ladder is leaning against a wall with its base 2 m from the wall. A shorter 2.4 m ladder is placed against the wall parallel to the longer ladder. • (a) How far will the smaller ladder reach up the wall? • (b) How far will the smaller ladder’s base be from the wall?

  10. 5.4 Modelling with Similar Triangles

  11. 5.4 Modelling with Similar Triangles

  12. 5.4 Modelling with Similar Triangles • AB is ll (parallel) to DE (ladder at same ∠) • ∠ABC = ∠DEC (F pattern) • ∠BAC = ∠EDC (F pattern) • ∠ACB = ∠DCE (common 90°) • Therefore, ∆ABC ~ ∆DEC

  13. 5.4 Modelling with Similar Triangles • For (a), we first need to find out how far the larger ladder reaches up the wall using the pythagorean theorem…

  14. 5.4 Modelling with Similar Triangles • c² = a² + b² (Pytharorean theorem) • DE² = DC² + EC² (Substitute triangle sides) • 3.6² = DC² + 2² (Use algebra to solve) • 12.96 = DC² + 4 • 12.96 – 4 = DC² • DC²=8.96 (Take √ of both sides) • √DC²= √8.96 • DC≈3 m

  15. 5.4 Modelling with Similar Triangles

  16. 5.4 Modelling with Similar Triangles Since ∆ABC ~ ∆DEC (a) to find AC (how far smaller ladder reaches up the wall)… • Plug in the known values • Cross-multiply • Simplify & solve for AC

  17. 5.4 Modelling with Similar Triangles • How far will the smaller ladder be from the base of the wall? • Here, we have EC, so we can set-up a ratio using our similar triangle relationship

  18. 5.4 Modelling with Similar Triangles Since ∆ABC ~ ∆DEC to find BC (how far smaller ladder base will be from the wall)… • Plug in the known values • Cross-multiply • Simplify & solve for BC

  19. Homework • P.474, #1-4 & 6-13

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