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Right Triangles Chapter 8: (page 284) Similarity in Right Triangles 8-1: (page 285) If a, b, and x are positive numbers and a : x = x : b , then x is the between a and b. Notice that x is both Example 1: in the proportion. Find the geometric mean between: (a) 12 and 3 (b) 7 and 14 (d) 64 and 49 (e) 1 and 3 Theorem 8-1 (c) (f) If the altitude is drawn to the 5 and 20 100 and 6 of a right triangle, then the two triangles formed are similar to the triangle and to each other. C Given: ∆ ABC with Rt. ∠ACB and with altitude CN Prove: ∆ ACB ~ ∆ ANC ~ ∆ CNB A N N A B N C C Proof: All three triangles can be proven similar by the B . Corollary 1 When the altitude is drawn to the of a right triangle, the length of the altitude is the between the segments of the hypotenuse. Given: !ABC with Rt.!ACB and with altitude CN C Prove: AN CN = CN BN A Proof: From Theorem 8-1, ∆ ACN ~ ∆ N B , therefore, this can be proven because corresponding sides of similar triangles are in Corollary 2 . When the altitude is drawn to the of a right triangle, each leg is the between the hypotenuse and the segment of the hypotenuse that is adjacent to that leg. Given: !ABC with Rt.!ACB and with altitude CN C Prove: (1) AB AC = AC AN (2) AB BC = BC BN Proof: (1) From Theorem 8-1, ∆ ABC ~ ∆ (2) From Theorem 8-1, ∆ ABC ~ ∆ A N B , and , therefore, (1) and (2) can be proven because corresponding sides of similar triangles are in . Informal statements of the corollaries: Y X A Z XA YA = YA AZ piece of hypotenuse altitude = altitude other piece of hypotenuse For leg XY : XZ XY = XY XA hypotenuse leg = leg piece of hypotenuse adjacent to leg For leg YZ : XZ YZ = YZ AZ hypotenuse leg = leg piece of hypotenuse adjacent to leg Corollary 1 Corollary 2 Example 2: (a) If CN=8 & NB=16, then AN = C A N B . (b) If AN=8 & NB=12, then CN = . C A (c) N B If AC=6 & AN=4, then NB = . C A (d) N B If CB=20 & NB=16, then AB = . C A N B Assignment: Written Exercises, pages 288 & 289: 17-41 odd #’s The Pythagorean Theorem 8-2: (page 290) One of the best known and most useful theorems in all of mathematics is the Theorem. This was named after , a Greek mathematician and philosopher. Many proofs of this theorem exist, including one by President . Pythagorean Theorem Theorem 8-2 In a right triangle, the of the hypotenuse is equal to the sum of the of the legs. C Given: ∆ ABC ; ∠ACB is a right angle Prove: Proof: A Statements B Reasons 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. 7. 7. ALSO: See #41 on page 289 and the Challenge Exercise on page 294. Examples: Find the value of “x”. (1) (2) 5 4 x x 2 5 12 (3) (4) 6 x 13 13 x 9 |----------- 10 ----------| (5) A B x (6) 5 x 4 3 D C |----------------- 17 ------------------| AC=12; BD=16 Assignment: Written Exercises, pages 292 & 293: 1-31 odd, 33-36 Prepare for Quiz on Lessons 8-1 & 8-2: Right Triangles The Converse of the Pythagorean Theorem 8-3: Theorem 8-3 If the (page 295) of one side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a triangle. C Given ∆ ABC with Prove: ∆ ABC is a triangle. A Example 1: (a) B Are the given lengths sides of right triangles? 4, 7, 9 (b) 20, 21, 29 (c) 0.8, 1.5, 1.7 A triangle with sides of 3, 4, and 5 is a right triangle because This is a very common triangle, called a - - . triangle. Any triangle with sides 3n, 4n, and 5n , where n > 0, is also a right triangle, because . Multiples of any 3 lengths that form a right triangle will also form triangles. These groups of 3 lengths are called . Some Common Right Triangle Lengths: 3, 4, 5 5, 12, 13 8, 15, 17 7, 24, 25 ____________ ____________ ____________ ____________ ____________ ____________ ____________ ____________ ____________ ____________ ____________ ____________ Theorem 8-4 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 C b a A If < Theorem 8-5 triangle. c B , then m∠C < 90º, and ∆ ABC is acute. + 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 C b a A c If Example 2: > triangle. B , then m∠C > 90º, and ∆ ABC is obtuse. + If a triangle is formed with the given lengths, is it acute, right, or obtuse? (1) 8, 12, 13 (2) 4, 4, 7 (3) 8, 9, 12 (4) 8, 11, 15 (5) 4, 5, 6 (6) 8, 9, 17 Assignment: Written Exercises, page 297: 1-17 odd #’s Special Right Triangles 8-4: Theorem 8-6 In a 45º-45º-90º ∆ , the hypotenuse is (page 300) times as long as a leg. 45º _____ a 45º a Note: A 45º-45º-90º triangle is an isosceles right triangle with congruent Example 1: (a) (b) Find the length of the legs or the hypotenuse of each 45º-45º-90º triangle. If the legs = 5, then the hypotenuse = (b) . If the legs = 5 6 , (d) then hypotenuse = (e) . . If the hypotenuse = 10, then the legs = . If the hypotenuse = 8 2 , then the legs = (f) . If the legs = 3 2 , then the hypotenuse = . If the hypotenuse = 4 3 , then the legs = . In a 30º-60º-90º triangle, Theorem 8-7 the hypotenuse is the longer leg is as long as the shorter leg, and times as long as the shorter leg. A shorter leg = 30º c hypotenuse = = longer leg = b = 60º B a The shorter leg is opposite the Example 2: C º angle and the longer leg is opposite the º angle. Using the side given, find the other 2 sides of each 30º-60º-90º triangle. (a) shorter leg = 2 hypotenuse = longer leg = (b) hypotenuse = 12 shorter leg = longer leg = (c) longer leg = 6 shorter leg = hypotenuse = Assignment: Written Exercises, pages 302 & 303: 1-19 odd #’s, 20-29 ALL #’s, BONUS: 32 & 36 Prepare for Quiz on Lessons 8-3 & 8-4 The Tangent Ratio 8-5: (page 305) Trigonometry, comes from 2 Greek words, which mean “ ”. Our study of trigonometry will be limited to trigonometry. The tangent ratio is the ratio of the lengths of the . B A C Definition: tangent of !A = tan A = length of the leg opposite !A BC = = ________ length of the leg adjacent to !A AC tangent of !B = tan B = length of the leg opposite !B AC = = ________ length of the leg adjacent to !B BC opposite tan = ! ___ ___ ___ adjacent Example 1: Express tan A and tan B as ratios. B (a) tan A = (b) tan B = 17 C 15 A The table on page 311 gives approximate decimal values of the tangent ratio for some angles. Example 2: (a) tan 20º ≈ (b) tan 87º ≈ “≈” means “is approximately equal to” NOTE: Many calculators also give approximations of the tangent ratio. The table can also be used to find an angle measure given a tangent value. Example 3: (a) tan ≈ .5774 (b) tan ≈ 4.0108 Many calculators use inverse keys to get values for angle measures for a tangent value. Example 4: Find the value of x to the nearest tenth and y to the nearest degree. (a) (b) x 3 x 72º 37º 25 x≈ x≈ (c) (d) x yº 5 5 89 yº 4 y≈ y≈ Assignment: Written Exercises, pages 308 & 309: 1-27 odd #’s 8-6: The Sine and Cosine Ratios Review : The tangent ratio is the ratio of the lengths of the (page 312) . The sine ratio and cosine ratio relate the legs to the . B A C Definitions: sine of !A = sin A = length of the leg opposite !A BC = = ________ length of the hypotenuse AB length of the leg adjacent to !A AC cosine of !A = cos A = = = ________ length of the hypotenuse AB Review : tangent of !A = tan A = length of the leg opposite !A BC = = ________ length of the leg adjacent to !A AC A way to always remember the definitions (Note: these are NOT the definitions!): sin = opposite hypotenuse _______ _______ _______ cos = adjacent hypotenuse _______ _______ _______ tan = opposite adjacent _______ _______ _______ Express the sine and cosine of ∠ A & ∠ B as ratios. Example 1: B (a) sin A = (b) cos A = 13 (c) sin B = (d) cos B = C Example 2: A Use the trigonometry table or a calculator to find the values. (a) sin 22º ≈ (c) sin Example 3: 12 ≈ 0.8746 (b) cos 79º ≈ (d) cos ≈ 0.7771 Find the value of “x” & “y” to the nearest integer. (a) (b) 84 x x x y 38º y 55º |------------- 14 --------------| x ≈ ________ y ≈ ________ x ≈ ________ y ≈ ________ Example 4: Find “n” to the nearest degree. (a) nº 14 20 n≈ (b) Find the measures of the three angles of a 3-4-5 triangle. Assignment: Written Exercises, pages 314 to 316: 1-23 odd #’s Worksheet on Lessons 8-5 & 8-6: The Sine, Cosine, and Tangent Ratios 8-7: Applications of Right Triangle Trigonometry (page 317) horizontal ------------------------------------------------------- A 1 2 B The angles formed between the horizontal and the line of are the: ANGLE of DEPRESSION: when a point B is viewed from a higher point A, ie. ANGLE of ELEVATION: when a point A is viewed from a lower point B, ie. Examples 1: At a certain time, a post 6 ft tall casts a 3 ft shadow. What is the angle of elevation of the sun? Examples 2: A person in a lighthouse 22 m above sea level sights a buoy in the water. If the angle of depression to the buoy is 25º, how far from the base of the lighthouse is the buoy? Assignment: Written Exercises, pages 318 to 320: 1-13 odd #’s Worksheet on Lesson 8-7: Applications of Trigonometry Prepare for Quiz on Lessons 8-5 to 8-7: Trigonometry Prepare for Test on Chapter 8: Right Triangles .