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Geometry 7-1 Geometric Mean and the Pythagorean Theorem A

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Geometry 7-1 Geometric Mean and the Pythagorean Theorem A Geometry 7-1 Geometric Mean and the Pythagorean Theorem A. Geometric Mean 1. Def: The geometric mean between two positive numbers a and b is the a x positive number x where: = . x b Ex 1: Find the geometric mean between the $8,000 question and the $32,000 question on “Who Wants to be a Millionaire?”. Ex 2: Find the geometric mean between 2 and 10. B. Theorem 7-1 If the altitude is drawn from the vertex of the right angle of a triangle to its hypotenuse, then the two triangles formed are similar to the given triangle and to each other. A D VABC : VA___ : VB ___ B C C. Theorem 7-2 The measures of the altitude drawn from the vertex of the right angle of a right triangle to its hypotenuse is, the geometric mean between the measures of the two segments of the hypotenuse. A VADB : VBDC w D AD BD = DB DC x a w a = a x B C Ex 3: Find the length of the altitude, if the following is true. 20 6 D. Theorem 7-3 If the altitude is drawn to the hypotenuse of a right triangle, then the measure of a leg of the triangle is the geometric mean between the measures of the hypotenuse and the segment of the hypotenuse adjacent to that leg. A w VABC : VADB : VBDC D AD AB = b x AB AC a w b = B C b x + w (hyp) Ex 4: Find the length of the given sides if the following is true. 4 y x 6 z HW: Geometry 7-1 p. 346-348 13-32 all, 35-38 all, 42-43, 49-50, 55-65 odd Hon: 34, 44, Geometry 7-2 The Pythagorean Theorem and its Converse A. Theorem 7-4 - Pythagorean Theorem In a right triangle, the sum of the squares of the measures of the legs equals the square of the measure of the hypotenuse. c a2 + b2 = c2 a b Ex 1: Find x. 14 7 x B. Theorem 7-5 - Converse of the Pythagorean Theorem -If the sum of the squares of the measures of two sides of a triangle equals the square of the measure of the longest side, then the triangle is a right triangle. 1. A Pythagorean Triple is ___________ whole numbers that satisfy the equation a2 + b2 = c2 . Ex 2: Determine if the measures of these sides are the sides of a right triangle. 40, 41, 48 HW: Geometry 7-2 p. 354-356 13-29 odd, 30-35, 40, 46-47, 51-55 odd, 61-69 odd Hon: 39 Geometry 7-3 Special Right Triangles A. 45o − 45o − 90o Triangles Do the Pythagorean Theorem (solve for d) d 2 2 2 x a + b = c x2 + x2 = d 2 x 1. Theorem 7-6 - In a 45o − 45o − 90o triangle, the length of the hypotenuse is 2 times as s s 2 long as a leg. leg 2 = hypotenuse s Example 1: Find the length of the sides of the triangle. 6 Example 2: If the leg of a 45o − 45o − 90o triangle is 12 units, find the length of the hypotenuse. B. 30o − 60o − 90o Triangles 1. What is the relationship between the short leg C of a 30o − 60o − 90o, triangle and the hypotenuse? short leg (___) = hypotenuse a 2. Let’s do Pythagorean Theorem to solve for a. o o 2 2 2 60 60 a + x = (2x) A B x D x short leg (____) = long leg 1. Theorem 7-7 - In a 30o − 60o − 90o triangle, the length of the hypotenuse is twice the length of the o 30 short leg, and the length of the long leg is 3 times the length of the short leg. n 3 2n 60o n Example 3: Find AB and AC. B o 60 12 A C Example 4: VWXY is a 30o − 60o − 90o triangle with right angle X and WX as the longer leg. Graph points X (-2, 7) and Y(-7, 7), and locate point W in quadrant III. HW: Geometry 7-3 p. 360-362 12-25, 27, 29, 36, 37, 40, 43-44, 45-65 odd Hon: 26, 38 7-4 Trigonometry Ratios in Right Triangles A. Ratios 1. Trigonometry helps us solve measures in right triangles. a. Trigon means triangle b. Metron means measure B. Triangle measures B Abbreviation Definition sin A leg opposite ∠A a = hypotenuse c c cos A leg adjacent to ∠A b a = hypotenuse c tan A leg opposite to ∠A a = A C adjacent b b Example 1: Find the sin S, cos S, tan S, sin E, cos E, tan E. M 6 8 S 10 E Ex 2: Solve the triangle A 6 o 35 C B Ex. 3: Solve the triangle 10 X Y 4 Y Ex 4: A plane is one mile above sea level when it begins to climb at a constant angle of 2o for the next 70 ground miles. How far above sea level is the plane after its climb? 1 mile HW: Geometry 7-4 p. 368-370 18-48, 63-64, 69-81 odd Hon: 55-58, 65-68 Geometry 7-5 Angles of Elevation and Depression A. Definitions: 1. An angle of elevation is the angle where if you start horizontal and move upward. Angle of elevation 2. An angle of depression is the angle where you start horizontal and move downward. Angle of depression Ex 1: A man stands on a building and sees his friend on the ground. If the building is 70 m tall and the angle of depression is 35o, how far is the man from the building? Ex 2: A man notices the angle of elevation to the top of a tree is 60o , if he is 14 m from the tree, how tall is the tree? HW: Geometry 7-5 p. 374-376 8, 9, 11, 13, 14-18, 28-29, 31-35 odd, 36-39, 41-47 odd Hon: 19, 24 Geometry 7-6 The Law of Sines A. The Law of Sines - In trigonometry, the Law of Sines can be used to find missing parts of triangles that are not right triangles. C 1. Let VABC be any triangle with sides a, b, and c representing the measures of the sides opposite the angles with measures A, B and C respectively. b a sin A sin B sinC Then = = . a b c A c B C 2. Proof of Law of Sines b Given: CD is an altitude of VABC . h a sin A sin B Prove: = a b A B D Statements Reasons 1.) CD is an altitude of VABC 1.)_____________ 2.) VACD and VCBD are rt V’s. 2.) Def of rt V’s. h h 3.) sin A = and sin B = 3.) Def of sine b a 4.) b(sin A) = h and h = a(sin B) 4.) ______________ 5.) b(sin A) = a(sin B) 5.)_______________ sin A sin B 6.) = 6.) Multiply each side by ____ a b Example 1: Find p. Round to the nearest tenth. Q 8 o o 17 29 P R Example 2: Solve VDEF if m∠D =112o, m∠F = 8o, and f = 2 Round to the nearest tenth. HW: Geometry 7-6 p. 381-383 17-35 odd, 38-39, 43, 46-58 Hon: 44-45 Geometry 7-7 The Law of Cosines A. The Law of Cosines - The Law of Cosines allows us to solve a triangle when the Law of Sines cannot be used. 1. Let VABC be any triangle with sides a, b, and C c representing the measures of the sides opposite the angles with measures A, B and C b respectively. Then the following equations a are true: 2 2 2 a = b + c − 2bccos A A c B b2 = a2 + c2 − 2accos B c2 = a2 + b2 − 2abcosC 2. You can use the Law of Cosines when you know two sides and the included angle. C o 3Example 1: Find c if b = 8, a = 6, and ∠C + 48 o 2 2 2 48 c = a + b − 2abcosC 8 6 A c B 3. You can use the Law of Cosines when you know all three sides and are looking for an angle. Example 2: Use the Law of Cosines to solve for ∠A. A a2 = b2 + c2 − 2bccos A 8 10 B C 12 HW: Geometry 7-7 p. 388-390 11-37 odd, 42, 46-47, 49-53 odd Hon: 39, 43, 57, 59 .
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