Chapter 8: Right Triangles (page 284)

Transcription

1 hapter 8: Right Triangles (page 284) 8-1: Similarity in Right Triangles (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 in the proportion. Example 1: Find the geometric mean between: (a) 12 and 3 (b) 7 and 14 (c) 5 and 20 (d) 64 and 49 (e) 1 and 3 (f) 100 and 6 Theorem 8-1 If the altitude is drawn to the of a right triangle, then the two triangles formed are similar to the triangle and to each other. Given: with Rt. and with altitude N Prove: ~ N ~ N N N N Proof: ll three triangles can be proven similar by the.

2 orollary 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:! with Rt.! and with altitude N Prove: N N = N N N Proof: From Theorem 8-1, N ~, therefore, this can be proven because corresponding sides of similar triangles are in. orollary 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:! with Rt.! and with altitude N Prove: (1) (2) = N = N N Proof: (1) From Theorem 8-1, ~, and (2) From Theorem 8-1, ~, therefore, (1) and (2) can be proven because corresponding sides of similar triangles are in.

3 Informal statements of the corollaries: Y X Z orollary 1 X Y = Y Z piece of hypotenuse altitude = altitude other piece of hypotenuse orollary 2 For leg XY : XZ XY = XY X hypotenuse leg = leg piece of hypotenuse adjacent to leg For leg YZ : XZ YZ = YZ Z hypotenuse leg = leg piece of hypotenuse adjacent to leg Example 2: (a) If N=8 & N=16, then N =. N

4 (b) If N=8 & N=12, then N =. N (c) If =6 & N=4, then N =. N (d) If =20 & N=16, then =. N ssignment: Written Exercises, pages 288 & 289: odd # s

5 8-2: The Pythagorean Theorem (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. Theorem 8-2 Pythagorean Theorem In a right triangle, the of the hypotenuse is equal to the sum of the of the legs. Given: ; is a right angle Prove: Proof: Statements Reasons LSO: See #41 on page 289 and the hallenge Exercise on page 294.

6 Examples: Find the value of x. (1) (2) 4 5 x x (3) (4) 6 x x (5) (6) 5 x x 4 3 D =12; D=16 ssignment: Written Exercises, pages 292 & 293: 1-31 odd, Prepare for Quiz on Lessons 8-1 & 8-2: Right Triangles

7 8-3: The onverse of the Pythagorean Theorem (page 295) Theorem 8-3 If the 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. Given with Prove: is a triangle. Example 1: re the given lengths sides of right triangles? (a) 4, 7, 9 (b) 20, 21, 29 (c) 0.8, 1.5, 1.7 triangle with sides of 3, 4, and 5 is a right triangle because. This is a very common triangle, called a - - triangle. ny 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 ommon Right Triangle Lengths: 3, 4, 5 5, 12, 13 8, 15, 17 7, 24, 25

8 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 triangle. b a c If < +, then m < 90º, and is acute. Theorem 8-5 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 triangle. b a c If > +, then m > 90º, and is obtuse. Example 2: 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 ssignment: Written Exercises, page 297: 1-17 odd # s

9 8-4: Special Right Triangles (page 300) Theorem 8-6 In a 45º-45º-90º, the hypotenuse is times as long as a leg. 45º a 45º a Note: 45º-45º-90º triangle is an isosceles right triangle with congruent. Example 1: Find the length of the legs or the hypotenuse of each 45º-45º-90º triangle. (a) If the legs = 5, (b) If the legs = 3 2, then the hypotenuse =. then the hypotenuse =. (b) If the legs = 5 6, (d) If the hypotenuse = 8 2, then hypotenuse =. then the legs =. (e) If the hypotenuse = 10, (f) If the hypotenuse = 4 3, then the legs =. then the legs =.

10 Theorem 8-7 In a 30º-60º-90º triangle, the hypotenuse is the longer leg is as long as the shorter leg, and times as long as the shorter leg. shorter leg = 30º hypotenuse = = c b longer leg = = 60º a The shorter leg is opposite the º angle and the longer leg is opposite the º angle. Example 2: 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 = ssignment: Written Exercises, pages 302 & 303: 1-19 odd # s, LL # s, ONUS: 32 & 36 Prepare for Quiz on Lessons 8-3 & 8-4

11 8-5: The Tangent Ratio (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. Definition: tangent of! = tan = tangent of! = tan = length of the leg opposite! length of the leg adjacent to! = = length of the leg opposite! length of the leg adjacent to! = = tan = opposite adjacent! Example 1: Express tan and tan as ratios. (a) tan = 17 (b) tan = 15

12 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 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 x 72º º x x (c) (d) x yº yº y y ssignment: Written Exercises, pages 308 & 309: 1-27 odd # s

13 8-6: The Sine and osine Ratios (page 312) Review : The tangent ratio is the ratio of the lengths of the. The sine ratio and cosine ratio relate the legs to the. Definitions: sine of! = sin = length of the leg opposite! length of the hypotenuse = = cosine of! = cos = length of the leg adjacent to! length of the hypotenuse = = Review : tangent of! = tan = length of the leg opposite! length of the leg adjacent to! = = way to always remember the definitions (Note: these are NOT the definitions!): sin = opposite hypotenuse cos = adjacent hypotenuse tan = opposite adjacent

14 Example 1: Express the sine and cosine of & as ratios. (a) sin = (b) cos = (c) sin = (d) cos = Example 2: Use the trigonometry table or a calculator to find the values. (a) sin 22º (b) cos 79º (c) sin (d) cos Example 3: Find the value of x & y to the nearest integer. (a) (b) 84 x x x y 38º 55º y x y x y

15 Example 4: Find n to the nearest degree. (a) nº n (b) Find the measures of the three angles of a triangle. ssignment: Written Exercises, pages 314 to 316: 1-23 odd # s Worksheet on Lessons 8-5 & 8-6: The Sine, osine, and Tangent Ratios

16 8-7: pplications of Right Triangle Trigonometry (page 317) horizontal The angles formed between the horizontal and the line of are the: NGLE of DEPRESSION: when a point is viewed from a higher point, ie. NGLE of ELEVTION: when a point is viewed from a lower point, ie.. Examples 1: t a certain time, a post 6 ft tall casts a 3 ft shadow. What is the angle of elevation of the sun? Examples 2: 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? ssignment: Written Exercises, pages 318 to 320: 1-13 odd # s Worksheet on Lesson 8-7: pplications of Trigonometry Prepare for Quiz on Lessons 8-5 to 8-7: Trigonometry Prepare for Test on hapter 8: Right Triangles