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# Geometry/Trigonometry Unit 7: Right Triangle Notes (1) Page 430 #1 – 15 (2) Page 430 – 431 #16 – 23, 25 – 27, 29 and 31 (3) Page 437 – 438 #1 – 8, 9 – 19 odd (4) Page 437 – 439 #10 – 20 Even, 23, and 29 (5) Page 444 #1 – 14 (6) Page 444 – 445 #15 – 26, 30 – 33 (7) Page 451 #1 – 8 (8) Page 451 – 452 #9 – 20 (9) Page 457 #1 – 10 Name: Date: Period: (10) Page 457 – 458 #11 – 28 (11) Page 465 #1 – 18 (12) Page 465 – 466 #19 – 29 Odd, 34 – 41 all (13) Page 469 – 471 #1 – 8, 12, 13, 15, 16, 18 – 24 Even, 29 – 32, 39 – 44 (14) Page 473 #1 – 21 Geometry Notes 9.1 Exploring Right Triangles All right triangles are either __________________________ To prove that two right triangles are similar you can show that _____________________________ of one of the triangles is _________________________ to______________ of the ___________________ of the other triangle. Theorem 9.1 (Use 3x5 Card to Explore): If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other. Therefore: . What triangles are similar? Theorem 9.2: In a right triangle, the length of the altitude from the right angle to the hypotenuse is the geometric mean of the lengths of the two segments of the hypotenuse. Theorem 9.3: In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. Each leg of the right triangle is the geometric mean of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg. and Geometry Notes 9.2 The Pythagorean Theorem Theorem 9.4 The Pythagorean Theorem: In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. A set of three positive integers, a, b and c, that satisfy the equation is a Pythagorean Triple. (ex1) 3, 4, 5 (ex2) _____________________ (ex3) __________________ (ex4) 5, 12, 13 (ex5) ______________________ (ex6) __________________ (ex7) 7, 24, 25 (ex8) ______________________ (ex9) __________________ Geometry Notes 9.3 The Converse of the Pythagorean Theorem Theorem 9.5 Converse of the Pythagorean Theorem: If the square of the length of the longest side of a triangle is equal to the sum of the squares of the two shorter sides, then the triangle is a right triangle. Theorem 9.6: If the square of the length of the longest side of a triangle is less than the sum of the squares of the lengths of the two shorter sides, then the triangle is acute. If a2 + b2 > c2 or c2 < a2 + b2, then the triangle is an acute triangle Theorem 9.7: If the square of the length of the longest side of a triangle is greater than the sum of the squares of the lengths of the two shorter sides, then the triangle is obtuse. If a2 + b2 < c2 or c2 > a2 + b2, then the triangle is an obtuse triangle Geometry Notes 9.4 Special Right Triangles 45-45-90 Theorem 9.8 – In a 45-45-90 triangle, the hypotenuse is √ times as long as each leg. That is, the sidelength ratios of leg:leg:hyp are 1:1:√ . E1. Find the missing values E2. Find the missing values P1. Find the missing values P2. Find the missing values 30-60-90 Theorem 9.9 – In a 30-60-90 triangle, the hypotenuse is twice as long as the shorter leg and the longer leg is √ as long as the shorter leg. That is, the side-length ratios of short leg : long leg : hyp is 1:√ :2 E1. Find t he missing values E2. Find the missing values E3. Find the missing values P1. Find the missing values P2. Find the missing values P3. Find the missing values Geometry Notes 9.5 Trigonometric Ratios _______________________________ – measurement of triangles From <A: From <B: B B _______________ c_______ _______________ a _______ _________________ c ____ _______________ a _______ C A C b _____________________ A _______________ b _______ Trigonometric Ratio – a ratio of the lengths of two sides of a triangle. Sine (sin), Cosine (cos) and Tangent (tan) are the_______ basic ______________________ ratios Ex1. Find each trigonometric ratio: (a) sin A= _______ (b) cos A=_______ (c) tan A=_______ (d) sin B=_______ (e) cos B=_______ (f) tan B=_______ B 5 C B 5 13 A 12 13 C 12 Find the missing side or angle and round to the nearest tenth E2. E3. E4. E5. A Geometry Notes 9.6 Extended Application: Solving Right Triangles: Solving a right triangle means to find the measure of __________________________ of the triangle and __________________________ of the triangle. In other words all __________parts. You can solve a right triangle if you know: A. B. (1) Two __________ lengths OR (2) One ____________________ length and one ______________________ measure How to find an angle measure of a right triangle given two side measures E1. Sin A = .2548 E2. Cos A = .5624 E3. Tan A = P1. Sin A = .6895 P2. Cos A = .4156 P3. Tan A = How to solve a right triangle given two side lengths E4. B P4. 13 C C. B 5 2 A 12 C A How to solve a right triangle given one side length and an acute angle measure E5. P5. B B 3 C ° 7 A C 54° A Geometry Notes Law of Sines and Law of Cosines: E1. Solve the triangle (AAS) E2. Solve the triangle (ASA) E3. Solve the triangle (SSA – One Triangle) – The ambiguous case E4. Solve the triangle (SSA – No Triangle) – The ambiguous case E5. Solve the triangle (SSA – Two Triangles) – The ambiguous case E6. Solve the triangle (SAS) E7. Solve the triangle (SSS)