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chapter 8 right triangles and trigonometry Section 8.2 The Pythagorean Theorem and Its Converse Pythagorean Theorem: In a right triangle, the sum of the squares of the legs equals the square of the hypotenuse. The legs are represented as a and b, the hypotenuse is represented as c. There are two shortcuts that can usually be used that involve the Pythagorean Theorem: To find the Hypotenuse (c): 1. square the legs 2. Add them together 3. Take the square root of the sum To find a Leg (a or b): 1. square the hypotenuse and the known leg 2. Subtract them 3. Take the square root of the difference Example: Example: Converse of the Pythagorean Theorem: If the sum of the squares of two sides equals the square of the third side, then the triangle is right. So if right. , then the triangle is Example: Is this a right triangle? Pythagorean Triples: three whole numbers that represent the sides of a right triangle. Some common examples of Pythagorean triples: 3-4-5, 6-8-10, 9-12-15, 30-40-50, 5-12-13, 8-1517, 7-24-25, 20-21-29 and there are many more examples Can these three lengths be the sides of a right triangle? Then state whether they form a Pythagorean triple. 10, 15, 20 5/8 9, 40, 41 3/8, 4/8, Area of Right Triangles Area = Base (one leg) * height (other leg) / 2 Section 8.3 Special Right Triangles 45-45-90 Triangle: OR Legs are congruent 30-60-90 Triangle: Examples: Other Examples: The perimeter of an equilateral triangle is 60 inches. Find the length of an altitude. The diagonal of a square is 14. Find the length of a side and then find the perimeter of the square. The altitude of an equilateral triangle is 10. Find the length of a side and then find the perimeter of the equilateral triangle. Section 8.4 Trigonometry Trig Ratios: SohCahToa Sine (sin): opposite leg / hypotenuse Cosine (cos): adjacent leg / hypotenuse Tangent (tan): opposite leg / adjacent leg Example: 10*sin38 = x Example: 10 * cos38 = y y = 10 / cos42 Example: 10 * tan42 = x y = 8 / sin64 x = 8 / tan64 Section 8.4 (part 2) Trig can be used to find the angle measures of a right triangle as well. Use the inverse of the sine or cosine or tangent to calculate the measure of an angle. Solve the Right Triangle (find all unknown measures) Section 8.5 Angles of Elevation and Depression Angle of Elevation: Angle between line of sight and horizontal when looking upward. Angle of Depression: Angle between line of sight and horizontal when looking downward. Example 1: The angle of elevation from point A to the top of a cliff is 34 degrees. If point A is 1000 feet from the base of the cliff, how high is the cliff? Example 2: The angle of depression from the top of an 80 foot building to point B on the ground is 42 degrees. How far from the base of the building is point B? Example 3: The tailgate of a truck is 3.5 feet above the ground. A loading ramp is attached to the tailgate with an incline of 10 degrees. Find the length of the ramp. Example 4: A sledding hill is 300 yards long with a vertical drop of 27.6 yards. Find the angle of depression of the hill. Percent Grade is calculated by dividing the rise (vertical elevation) by the run (horizontal distance) Example 5: What is the angle of elevation of these hills? Steepest street in the world 35% in New Zealand Green slopes are usually between 6% and 25% grade Blue slopes are between 25% and 40% Black slopes are greater than 40% What is the angle of depression range for each slope? Gage is standing on the ground 60 feet from a cliff looking up to the top with an angle of elevation of 500. His eye level is 6 feet above ground. How tall is the cliff? Now Gage is standing on top of a 40 foot building looking down at a car with an angle of depression of 120. His eye level is 6 feet above the top of the building. How far is it from the car to his eyes?