Right Triangles and Trigonometry: Geometry Chapter 9, Exercises of Trigonometry

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Right Triangles and

Trigonometry

Geometry

Chapter 9

• This Slideshow was developed to accompany the textbook
  • Big Ideas Geometry
  • By Larson and Boswell
  • 2022 K12 (National Geographic/Cengage)
• Some examples and diagrams are taken from the textbook.

Slides created by

Richard Wright, Andrews Academy

[email protected]

9.1 The Pythagorean Theorem

• Find the value of x
• Try # Pythagorean Theorem In a right triangle, a^2 + b^2 = c^2 where a and b are the length of the legs and c is the length of the hypotenuse. 3 ଶ^ + 𝑥ଶ^ = 5ଶ 9 + 𝑥ଶ^ = 25 𝑥ଶ^ = 16 𝑥 = 4 6 ଶ^ + 4ଶ^ = 𝑥ଶ 36 + 16 = 𝑥ଶ 52 = 𝑥ଶ 𝑥 = 2 13

9.1 The Pythagorean Theorem

• Pythagorean Triples
  • A set of three positive integers that satisfy the Pythagorean Theorem

9.1 The Pythagorean Theorem

• Show that the segments with lengths 3, 4, and 6 can form a triangle
• Classify the triangle as acute, right or obtuse.
• Try # If c is the longest side and… c^2 < a^2 + b^2  acute triangle c^2 = a^2 + b^2  right triangle c^2 > a^2 + b^2  obtuse triangle 3 + 4 > 6 7 > 6 3 ଶ^ + 4ଶ^? 6 ଶ 9 + 16? 36 25 < 36 obtuse

9.2 Special Right Triangles

After this lesson…

• I can find side lengths in 45°-45°-90° triangles.
• I can find side lengths in 30°-60°-90° triangles.
• I can use special right triangles to solve real-life problems.

9.2 Special Right Triangles

• 45 -45-90 • 30 -60-90 1 1 2 45° 45° 3 1 2 30° 60° If you have another 45-45-90 or 30°-60°-90° triangle, then use the fact that they are similar and use the proportional sides.

9.2 Special Right Triangles

• Find the value of x. Write your answer in simplest form.
• Try # 𝑥 22

9.3 Similar Right Triangles

After this lesson…

• I can explain the Right Triangle Similarity Theorem.
• I can find the geometric mean of two numbers.
• I can find missing dimensions in right triangles.

9.3 Similar Right Triangles

• ΔCBD ~ ΔABC, ΔACD ~ ΔABC, ΔCBD ~ ΔACD If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other. Right Triangle Similarity Theorem

9.3 Similar Right Triangles

• Find the geometric mean of 8 and 10.
• Try # The geometric mean of two positive numbers a and b is the positive number that satisfies ௔ ௫ = ௫ ௕ . So, 𝑥 = 𝑎𝑏 Geometric Mean 8 · 10 = 80 = 4 5 ≈ 8.

9.3 Similar Right Triangles

• 𝐶𝐷 = 𝐴𝐷 · 𝐷𝐵 If the altitude is drawn to the hypotenuse of a right triangle, then the altitude is the geometric mean of the two segments of the hypotenuse. Geometric Mean (Altitude) Theorem

9.3 Similar Right Triangles

• Find the value of x or y.
• Try # 𝑥 9

𝑥ଶ^ = 45

𝑦ଶ^ = 40

9.4 The Tangent Ratio

After this lesson…

• I can explain the tangent ratio.
• I can find tangent ratios.
• I can use tangent ratios to solve real-life problems.