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Geometry

Explore the geometric mean, Pythagorean Theorem, and triangle properties such as obtuse, acute, and 45-45-90 triangles. Learn about sine, cosine, and tangent ratios for right triangles.

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Geometry

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  1. Geometry Chapter 8 Review

  2. Geometric Mean Find the geometric mean between the two numbers. • 5 and 20 • 64 and 49

  3. Corollary 1 piece of hypotenuse altitude altitude other piece of hypotenuse = Y X A Z

  4. Corollary 2 hypotenuse leg leg piece of hyp. adj. to leg = Y X A Z

  5. Corollary 2 hypotenuse leg leg piece of hyp. adj. to leg = Y X A Z

  6. Pythagorean Theorem • In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs. A c b C a B

  7. Theorem: Converse of the Pythagorean Theorem If the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle. If c² = a² + b² Rt. ∆ c b a

  8. Theorem If the square of the longest side of a triangle is greater than the sum of the squares of the other two sides, then the triangle is an obtuse triangle. If c² > a² + b² Obtuse ∆ c b obtuse a

  9. Theorem If the square of the longest side of a triangle is less than the sum of the squares of the other two sides, then the triangle is an acute triangle. If c² < a² + b² Acute ∆ c b acute a

  10. 45-45-90 Triangles The formula. o 45 x o 45 x Since all 45-45-90 triangles are similar, by AA Similarity Postulate, this formula works for all 45-45-90 triangles.

  11. 30-60-90 Triangles The formula. o 30 2x o 60 x Since all 30-60-90 triangles are similar, by AA Similarity Postulate, this formula works for all 30-60-90 triangles.

  12. Tangent ratio= The tangent ratio is the ratio of the length of the legs in a Rt. ∆ opposite leg hypotenuse Tangent of <A: A adjacent leg

  13. Sine and Cosine Ratios The sine ratio is the ratio of the length of the legs in a Rt. ∆ opposite leg hypotenuse Sine of <A: A adjacent leg opposite leg hypotenuse Cosine of <A: A adjacent leg

  14. HW • W.S. Let’s do the odds on Chapter 8 side together!

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