Name Date Period Notes Formal Geometry Chapter 8 Right Triangles and Trigonometry 8.1 Geometric Mean. A. Definitions: 1.

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1 Name Date Period Notes Formal Geometry Chapter 8 Right Triangles and Trigonometry 8.1 Geometric Mean A. Definitions: 1. Geometric Mean: 2. Right Triangle Altitude Similarity Theorem: If the altitude is drawn to the of a right triangle, then the two triangles formed are similar to the original triangle and each other. Example: Let s Explore: What do you notice about the circled proportions? 1

2 3. Geometric Mean (Altitude) Theorem: The altitude drawn to the hypotenuse of a right triangle separates the hypotenuse into two segments. The length of this altitude is the between the lengths of these two segments. Example: 4. Geometric Mean (Leg) Theorem: The altitude drawn to the of a right triangle separates the hypotenuse into two segments. The length of a leg of this triangle is the geometric mean between the length of the hypotenuse and the segment of the hypotenuse to that leg. Example: B. Examples: 1. Find the geometric mean between 8 and Find the geometric mean between 5 and Write a statement identifying the three similar right triangles in the figure. 4. Find the value for x, y and z. 2

3 5. Solve each proportion: 8.2 The Pythagorean Theorem and Its Converse A. Definitions: 1. Pythagorean Theorem: In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. Example: Prove the Pythagorean Theorem using Geometric Means: Given: ABC is a right triangle with legs of length a and b and hypotenuse of length c. Prove: a 2 + b 2 = c 2 Part I. Prove triangles similar: Draw a perpendicular from C to the hypotenuse. Label the point of intersection X. 1) BXC BCA because 2) B B by 3) So, BXC~ BCA by 4) AXC ACB because 5) A A by 6) So, AXC~ ACB by 3

4 Part II. Write proportions using Geometric Mean: Label AX = d and label XB = e. Use the fact that corresponding sides of similar triangles are proportional to write two proportions. Proportion 1: Proportion 2: Part II. Set up the equations and solve. Start with getting a and b by itself in each proportion. (you will have a 2 and b 2 ). Next add the two equations together. Use factoring and the fact that e + d = c to solve for the Pythagorean theorem. 2. Pythagorean Triple: 3. Converse of the Pythagorean Theorem: 4. Acute Pythagorean Inequality Theorem: If the square of the length of the side of a triangle is less than the sum of the squares of the lengths of the other two sides, then the triangle is a triangle. 5. Obtuse Pythagorean Inequality Theorem: If the square of the length of the longest side of a triangle is than the sum of the squares of the lengths of the other two sides, then the triangle is an triangle. 4

5 B. Examples: 1. Find the missing measures using the Pythagorean Theorem. 2. Use a Pythagorean triple to find each missing length. Explain your reasoning. A B 3. Df c. 5

6 8.3 Special Right Triangles Part 1 Part 2 6

7 A. Definitions: Triangle Theorem: Triangle Theorem: B. Examples: Find the missing side lengths. Leave your answers in simplest radical form

8 2. F Give exact and approximate (to the nearest tenth) answers. 5. a. Sketch an equilateral triangle. Sketch an altitude for your equilateral triangle. What do you notice? b. c. 6. a. Sketch a square. Sketch a diagonal of your square. What do you notice? b. c. 8

9 8.4 Trigonometry A. Opener: 1. Write the equation of a line parallel to y = 3x 10 that passes through (2, -1). B. Definitions: 1. Trigonometry: 2. Trigonometric Ratio: a. Sine: (sin) b. Cosine: (cos) c. Tangent: (tan) 3. Inverse Trigonometric Ratio: C. Examples: 1. Express each ratio as a fraction and as a decimal rounded to the nearest hundredth. a) Sin P d) Sin Q b) Cos P e) Cos Q c) Tan P f) Tan Q *What do you notice about the sinp and cosq; cosp and sinq? Write a rule: 9

10 2. Kl (Exact and Approximate measures.) 2b. tan 60 degrees. 2c. cos 45 degrees. 3. Find x to the nearest hundredth. 4. df 10

11 5. Solve each right triangle. Round angle measures to the nearest degree and side measures to the nearest tenth. What does it mean to SOLVE a triangle? 8.5 Angles of Elevation and Depression A. Opener: Simplify. 1. (3xy 3 ) 2 2. (4x) 1 3. (x 5 )(x 2 ) 3 4. (5x) 0 B. Definitions: 1. Angle of elevation: 2. Angle of depression: C. Examples: kicked to the nearest degree? Draw a sketch!!! 11

12 the swimmer? to the nearest meter? 8.6 The Law of Sines A. Opener: 1. How tall is the tree? 2. How tall is this windblown tree? 12

13 Deriving the Law of Sines: ABC does not contain a right angle. An altitude is dropped from vertex B. Part 1 Step 1. sina = sinc = 2. Solve the equations above for h. 3. Since both equal h, now we can set them. 4. Rearrange your equation so it looks like a proportion. Part 2 - Drop an altitude from vertex A and repeat Part 1 for sinb and sin C. Part 3 What are your final conclusions for Parts 1 and 2? B. Definitions: 1. Law of Sines: 2. When will I use the Law of Sines? 13

14 C. Examples: Find the value of x. Round to the nearest tenth D. Guided Practice: Find the value of x. Round to the nearest tenth

15 E. Proof: Given: CD is an altitude of ABC sin A sin B Prove: = a b (Hint: think about our derivation of the Law of Sines!) 8.6 Continued: The Law of Cosines A. Deriving the Law of Cosines ABC does not contain a right angle. An altitude has been dropped from vertex B. Part 1 1. Define sina = and cosa = 2. Write sina in terms of h and cosa in terms of r. Part 2 Given: sin 2 A + cos 2 A = 1 (This is a trig identity ) 1. Write the Pythagorean Theorem with the hypotenuse on the left. 2. CD = 3. Now write the Pythagorean Theorem for CBD. 4. Substitute the expressions for h and r from Part Simplify the equation. 15

16 B. Definitions: 1. Law of Cosines: 2. When will I use the Law of Cosines? 3. Solve a Triangle: C. Examples: Find the measure indicated. Round to the nearest tenth. 16

17 D. Proof: Given: h is an altitude of ABC Prove: c 2 = a 2 + b 2 2ab cos C 17

18 OC 6.5 Deriving the Area Formula for any Triangle PART 1 18

19 Using the Area Formula PRACTICE 19