7.1/7.2 Apply the Pythagorean Theorem and its Converse

Transcription

1 7.1/7.2 Apply the Pythagorean Theorem and its Converse Remember what we know about a right triangle: In a right triangle, the square of the length of the is equal to the sum of the squares of the lengths of the. + = Remember that a and b represent the lengths of the legs and c always represents the length of the hypotenuse. There are many different methods to prove the Pythagorean Theorem. One method is to use the distance formula. (4, 7) c (1, 3) (4, 3) Find missing sides using the Pythagorean Theorem: Find the area of the following triangle.

2 3. A 5 foot board rests under a doorknob and the base of the board is 3.5 feet away from the bottom of the door. Approximately how high above the ground is the doorknob? Pythagorean Triple: Pythagorean Converse: Use the converse to determine if a triangle is, or. Triangle Inequality Theorem:

3 Classify the triangle as acute, obtuse, or right , 26, 14 6.

4 7.3 Similar Right Triangles Identify the similar triangles in the diagram. Sketch the three similar right triangles so that the corresponding angles and sides have the same orientation. Corresponding Sides of Similar Triangles are in Proportion EXAMPLES: 17: 20: y: Find the values of the missing variables after you decide which proportion to set up (leg or alt)

5 7.4 Special Right Triangles Radical Review: Simplify the following radicals. NO DECIMALS! A triangle is an isosceles right triangle that can be formed by cutting a square in half. Find the missing sides of each triangle

6 A triangle is a right triangle that can be formed by cutting an equilateral triangle in half. Find the missing sides of each triangle

7 7.5/7.6 APPLY THE TANGENT, SINE, AND COSINE RATIOS VOCABULARY TRIGONOMETRY: USED TO FIND MEASURES IN TRIANGLES TRIGONOMETRIC RATIO TANGENT RATIO SINE RATIO COSINE RATIO SOH CAH TOA When we define our trigonometric ratios, we label each side of a right triangle as opposite leg (opp), adjacent leg (adj.), or hypotenuse with respect to which angle we are using. Sometimes we use the symbol theta,, in place of x for an angle. Do you remember Example: Label each side as opposite, adjacent, or hypotenuse with respect to. how to find a missing side of a right triangle? Find the three trig ratios for each angle. sin C = a 2 b 2 c sin B = 2 cos C = cos B = tan C = tan B =

8 A sin A = sin B = cos A = cos B = tan A = tan B = C B Use the trig functions to help find missing sides of a triangle. You can use your calculator to evaluate trig functions of any angle. Use the keys,. Before using the calculator, make sure it is set in DEGREE MODE.

9 Use the inverse trig functions to help find missing angles of a triangle. When trying to find a missing angle, use your trig inverse button by pressing the 2 ND button before. ( sin 1, cos 1, 1 and tan ) Use your calculator to estimate the measure of A to the nearest tenth of a degree. sin A = 0.76 cos A = 0.17 tan A = 0.9 Use a calculator to approximate the measure of A to the nearest tenth of a degree. Solve the right triangle. (Find all missing parts)

10 Angle of Elevation and Angle of Depression: VOCABULARY ANGLE OF ELEVATION If you look up at an object, the angle your line of sight makes with a horizontal line is called the angle of elevation. EXAMPLE: Railroad A railroad crossing arm that is 20 feet long is stuck with an angle of elevation of 35 Find the lengths x and y ANGLE OF DEPRESSION If you look down at an object, the angle your line of sight makes with the horizontal line is called the angle of depression. EXAMPLE: Roller Coaster You are at the top of a roller coaster 100 feet above the ground. The angle of depression is 44. About how far do you ride down the hill? Notice how the angle of angle of elevation is always equal to the angle of depression. Do you remember why from first semester? Solve the following story problems. Draw your own picture for each. You attend a music concert with some friends and sit You are standing 350 feet away from a skyscraper halfway up the bleachers in the arena. The angle of that is 750 feet tall. What is the angle of elevation depression from your horizontal line of sight to the from you to the top of the building? stage is 24. If your seat is 45 feet above stage level, what is your actual distance d from the stage?

11 Chapter 7 Extension Law of Sines and Law of Cosines Review: Right Triangles Non-right Triangles Pythagorean Theorem SUM OF ANGLES OF A TRIANGLE =180 Special Right Triangles ( ) and ( ) SOH CAH TOA *LAW OF SINES *LAW OF COSINES Notice that A and a B and b C and c Examples: 1) 2) x = x =

12 3) Solve the triangle. AC = b AB = B 4) Solve the triangle. AC = C = B b

13 Examples: 1) 2) x = x = 3) Solve the triangle. AC = A = C

14 4) Solve the triangle. B = C = A