hapter 7 Right Triangles and Trigonometry 7.1 pply the Pythagorean Theorem 7.2 Use the onverse of the Pythagorean Theorem 7.3 Use Similar Right Triangles 7.4 Special Right Triangles 7.5 pply the Tangent Ratio 7.6 pply the Sine and osine Ratios 7.7 Solve Right Triangles SOL G.7 The student, given information in the form of a figure or statement, will prove two triangles are similar, using algebraic or coordinate methods as well as deductive proofs. Name lock
7.1 and 7.2 Pythagorean Theorem Pythagorean Theorem a b c Pythagorean Theorem is used to find the third side of any. Sides a and b are called the. Side c is called the. For any right triangle,. Find the length of the unknown side of the right triangle. Determine whether the unknown side is a leg or a hypotenuse. 1. 2. 8 3 6 5 3. 4. 12 4 9 6 5. 16 foot ladder rests against the side of a house, and the base of the ladder is 4 feet away. pproimately how high above ground is the top of the ladder? 6. Find the area of an isosceles triangle with side lengths 10 meters, 13 meters, and 13 meters.
Tell whether the given triangle is a right triangle. 7. 8. 3 34 9 26 22 14 15 Tell whether a triangle with given side lengths is a right triangle. 9. 4, 4 3, and 8 10. 10, 11 and 14 11. 5, 6 and 61 lassifying Triangles Given a triangle with sides a, b and c If, then the triangle is a triangle. If, then the triangle is a triangle. If, then the triangle is a triangle. Keep c 2 on the left! 12. an segments with lengths of 4.3 feet, 5.2 feet, and 6.1 feet form a triangle? If so, would the triangle be acute, right or obtuse? 13. Show that segments with lengths 3, 4 and 6 can form a triangle and classify the triangle as acute, right or obtuse.
You try! In #1-6, use to determine if the equation is true or false. 1. b 2 + a 2 = c 2 4. c 2 = a 2 - b 2 b c 2. c 2 b 2 = a 2 5. c 2 = b 2 + a 2 a 3. b 2 c 2 = a 2 6. a 2 = c 2 b 2 Find the unknown side lengths. Radical answers should be written in simplest form. 7. 8. 9. 12 19 7 6 13 5 Find the area of the figure. Write answers in simplest radical form when necessary. 10. 11. 12. 10 ft 11 in 7 ft 14 in 4 cm 20 in Decide whether the numbers can represent the side lengths of a triangle. If they can, classify the triangle as right, acute or obtuse. 13. 5, 12 and 13 14. 8,4 and 6 15. 20, 21, and 28
7.3 Similar Right Triangles Video tutorials for Geometric Mean If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar the original and each other. ~, ~ and ~ ltitude Theorem Geometric Mean Leg Theorem Side 1 ltitude = ltitude Side 2 altitude ig hyp = ig leg Little leg Little hyp ig leg and little hypotenuse *hint look for the T shape* Find the geometric mean of the two numbers. 1. 4 and 25 2. 6 and 20 *hint look for the L shape* Solve for the variable. Leave all answers in simplest radical form. 3. 4. R y 3 S 9 10 2 k Q P
omplete and solve the proportions. 5. 6. 7. 15 = = 12 8 9 = 12 11 8 20 15 9 Find the value(s) of the variable(s). You try! 1. 2. 3. 4. Tell whether the triangle is a right triangle. 5. 6. 7.
Warm-Up: Discovering Special Right Triangles 1. What are the measures of and? Eplain how you arrived at your answer. 2. lassify the triangle above by its angles and sides. 3. Write the Pythagorean theorem. Your equations will all be in terms of and h because the legs of the triangle are units long and the hypotenuse is h units long. 4. Solve for h. Your answer should be in simplest radical form. (Think lgebra!!) 5. Redraw the triangle above, substituting your answer from step 3 for h in the diagram. 6. Summarize your findings in a complete sentence.
7.4 Special Right Triangles Video tutorial for special right triangles Review: Rationalizing the Denominator Remove the radical from the denominator by multiplying the numerator and the denominator by a fraction of 1. 1. 2 2. 15 3. 1 3 5 3 6 Special Right a a a 2 Hypotenuse = Leg = The legs of a 90 triangle are always congruent. Directions: Find the length of the hypotenuse. Write all radical answers in simplest form. (No decimals) 1. 2. 3. 8 3 2
Directions: Find the length of the legs in the triangle. Write all radical answers in simplest form. (No decimals) 4. 5. 6. 2 2 Hypotenuse = 3 30 2 Long leg = Special Right Long Leg 60 Short leg = 1 Short Leg Short leg = Directions: Find the values of the variables. Write all radical answers in simplest form. (No decimals) 7. 8 9. The logo on a recycling bin 60 resembles an equilateral triangle y 30 with side lengths of 6 cm. What is the approimate height of the 3 logo? 9
You try! Find the value of. Write your answer in simplest radical form. 1. 2. 3. 8 4 2 4 4. 5. 6. 10 9 2 Find the value of each variable. Write your answer in simplest radical form. 7. y 8. 9. 30 30 3 3 y 60 60 30 8 y 18 60
7.5 and 7.6 Trigonometry Ratios Video tutorial for trig ratios Trigonometric RTIOS b How to remember the ratios! a c Each acute angle of a right triangle has the following trigonometric ratios. SINE OSINE TNGENT The ratio of the leg Sin = the angles to the. Sin = The ratio of the leg os = to the angle to the. os = The ratio of the leg Tan = the angle to the leg to the Tan = angle. SOH H TO Sin = os = Tan = Directions: Find the sine, cosine, and tangent ratios for each acute angle in the triangle. Write each answer as a decimal. Round to the nearest thousandths place when necessary. 1. 2. 17 8 S 16 65 15 T 63 R sin : sin : sin : sin : cos : cos : cos : cos : tan : tan : tan : tan :
Find the value of each variable. Round to the nearest hundredths place. 3. 4. 5. 18 32 y 51 y 10 48 6. Jake leaned a 12 foot ladder against his house. If the angle formed by the ladder and the ground is 68, how far from the back of the house did he place the ladder? 7. ramp is used to load suitcases on an airplane. If the cargo door is 7 feet from the ground and the angle formed by the end of the ramp and the ground is 25, how long is the ramp? ngle of Elevation Looking up at an object, then angle your line of sight makes with a horizontal line Hey down there! ngle of depression ngle of Depression Looking down at an object, then angle your line of sight makes with a horizontal line ngle of elevation Hey up there! 8. asey sights the top of an 84 foot tall lighthouse at an angle of elevation of 58. If asey is 6 feet tall, how far is he standing from the base of the lighthouse?
9. lifeguard is sitting on a platform, looking down at a swimmer in the water. If the lifeguard s line of sight is 8 feet above the ground and the angle of depression to the swimmer is 18, how far away is the swimmer from the lifeguard? You try! Use a calculator to approimate the given value for four decimal places. 1. sin30 2. cos18 3. tan72 4. tan42 5. sin83 6. cos65 Find the value of each. Round decimals to the nearest tenth. 7. y 8. y 9. 36 10 64 8 y 48 8 10. pilot in a helicopter spots a landing pad below. If the angle of elevation is 73 and the horizontal distance to the pad is 1200 feet, what is the altitude of the helicopter? 11. The angle of elevation form a kicker s foot on the football field to the top of the goal post bars is 17. If he is standing 131 feet from the base of the goal post, how tall is the goal post?
7.7 Solving Right Triangles To solve a right triangle to find the measures of all the sides and angles you must know either: a. two side lengths b. one side length and one acute angle Inverse Trigonometric Ratios Inverse Sine Inverse osine If sin =, then sin -1 = m If cos = y, then cos -1 y = m Inverse Tangent If tan = z, then tan -1 z = m ** inverse ratios find the measure of the unknown NGLE ** 1. Use a calculator to approimate the measure of to the nearest tenth of a degree. a. b. 14 15 20 11 2. Let and be acute angles in a right triangle. Use a calculator to approimate the measures of and to the nearest tenth of a degree. a. sin = 0.87 b. cos = 0.15 3. Solve the right triangle. Round decimals to the nearest tenth. a. b. G 14 H 42 70 16 J
Laws for Solving Triangles Video tutorial for Law of Sines and Law of onsines Law of Sines This works for any non-right triangle when you are given sides and the angle OR two angles and the Law of osines This works for and non-right triangle when you are given two sides and the angle OR when you are given sides. side. Find the indicated measures. 1. Find. 2. Find. 3. Find m G. 4. For find the length of c given a = 14, b = 20, and m = 120.
You try! Find the indicated angle measures. 1. Find m. 2. Find m G. 3. Find m M. 8 G H 37 12 N 6 10 I O 26 M 4. Find m. 5. Find m T. 6. Find m D. 9 32 R S E 20 24 17 13 D 29 34 F T Find the indicated measures. 7. Find m T. 8. Find. 9. Find m. R 15 15 12 T 18 38 7 S 9 42 13