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Fun geometry Chapter 7 packet 7.1 note sheet Find the geometric mean between the two numbers: 1) 8 and 6 2) 4 and 9 3) 2 2 and 6 2 1 1) 2) 3) 4) Find x and/or y: 5) 6) x y x x = ______ x = ________, y = ________ 2 Leave answers in exact form, reduced. √6 and √6 Solve for each variable. Express answers in exact form. 15) ans wer s in exa ct for m, red 18) uce ans d.wer s in exa ct for m, red uce d. 16) ans wer s in exa ct for m, red19) uce ans d. wer s in exa ct for m, red uce d. 17) ans wer s in exa ct for m, red 20) uce ans d. wer s in exa ct for m, red uce d. 3 4 Geometry Notes Section 7.2: The Pythagorean Theorem and Its Converse The Pythagorean Theorem: In a right triangle, the sum of the squares of the __________ is equal to the ______________________________. Acute triangle: __________________________ Right triangle: __________________________ Obtuse triangle: _________________________ Pythagorean Triple: Three _________ numbers that can form sides of a right triangle. ______________, _______________, and ______________. Examples: 1) 2) 3) Determine if the following vertices form a right, obtuse, or acute triangle. Show all work and explain. 4) 11, 4, 10 5) 5, 6, 4 6) W(-3, 6) T(5, 5) Y(3, 2) 5 The Pythagorean Theorem Give all answers in exact form unless noted differently. 1) The sides of a rectangle are 7 inches long and 9 inches long. Find the length of the diagonal. 2) The diagonal in a square is 16 cm long. Find the length of the sides. 3) The base of an isosceles triangle is 8 ft long and the height is 3 ft. Find the length of the legs. 4) The perimeter of an equilateral triangle is 30 inches. Find the length of the altitude. 5) If a man travels east 3 miles and south 2.5 miles, approximately how far is he from his starting point? 6) A triangle has sides of 14, 18 and 15. Classify the triangle as acute, obtuse or right. 6 10) ans wer s in exa ct for 13) m, ans red wer uce s in d. exa ct for 11) ans wer s in exa ct for 14) m, ans red wer uce d.s in exa ct 12) ans wer s in exa ct for m,15) redans ucewer d. s in exa ct 7 7.1-7.2 practice 12. ans wer s in exa 13. ans wer s in exa 14. ans wer s in exa 15. ans wer s in exa 8 Do the following sides form an acute, right or obtuse triangle? Show work. Leave answers in exact form. 12 Decide if the following vertices form an acute, right or obtuse triangle. 9 Decide if the following vertices form an acute, right or obtuse triangle. Decide if the following sides of a triangle form an acute, right or obtuse triangle. Identify any Pythagorean triples. 10 45-45-90 Triangles Problems 1 to 3. Write the decimal value (rounded to the nearest tenth) 3 2 = 1) 2) 2 5 = 3) 5 2 = Problems 4 to 6. Solve for x and express it in simplest radical form. 4) 5) 6) x 3 4 x x Problems 7 to 18. Solve for x and express it in simplest radical form. 7) 8) 9) 17 x x 45 x 10) 11) 12) x x x 45 13) 14) 15) x 12 x 45 x 200 16) 17) 18) x 2 x x Problems 19 to 21. Simplify each radical. 19) 22) 4 2 20) 81 23) 2 2 24 21) 2 5 14 = 24) 5 80 = 3 7 5 11 Problems 25 to 42. Solve for x and express it in simplest radical form. 25) 26) 27) 9 x x x 28) 29) 30) x x 45 x 31) 32) 32 33) x x 45 x 34) 35) 36) x 30 38 x 45 x 37) 38) 39) x x x 40) 41) 42) x x 23 2 x 12 45-45-90 special right triangles Name: ____________________________________ 13 14 30-60-90 Triangles Rewrite each expression into simplest radical form. 1) 75 2) 3) 3 7 2 4) 18 2 20 10 Find x and y. Put the answer in simplest radical form. 5) x = _______, y = __________ 6) x = _______, y = __________ x 4 13 30 x 60 y y 7) x = _______, y = __________ 8) x = _______, y = __________ x 60 y 2 30 10 9) x = _______, y = __________ x 11) x = _______, y = __________ 10) x = _______, y = __________ 30 y 12) x = _______, y = __________ 18 30 y y x 30 y x x 30 y x 15 Find x and y. Put the answer in simplest radical form. 13) x = _______, y = __________ x 14) x = _______, y = __________ x 60 30 y y 15) x = _______, y = __________ 16) x = _______, y = __________ x 60 x y 30 17) x = _______, y = __________ y 18) x = _______, y = __________ 12 30 x y x 30 y 19) x = _______, y = __________ 20) x = _______, y = __________ 5 x 30 x y 30 y 21) x = _______ 1 22) x = _______, y = __________ 5 45 x y x 16 30-60-90 special right triangles 17 18 10) 11) 13) 14) 12) 15) 16) A square had a diagonal of 2√6 𝑖𝑛 find the perimeter. 17) An equilateral triangle has a perimeter of 30m, find the altitude. 19 20 21 Answer the following in exact form 13. 14. 15. 22 11. 12. 13. 14. 15. 16. 17. 23 7) 8) 9) 10) 11) 12) 13) 14) 24 7.4 Trigonometric ratios For any acute angle in a right triangle the following ratios are formed: Sine sin(angle) = _________________ Cosine cos (angle) = ________________ Tangent tan(angle) = _________________ Give the side adjacent <A ______________________ B Give the side opposite <A c ______________________ A Give the hypotenuse: ______________________ Give the side opposite <B ______________________ Give the side adjacent <B ______________________ a b C B 12 Give the following ratios rounded 4 decimal places A 11 C (hint: first find BC) 1) Sin<A = _________ 2) tan<A = _________ 3) Sin<B = _________ 4) cos<A = _________ 5) Tan<B = _________ 6) cos<B = _________ 25 Solving for right triangles using trigonometry Round all angles to the nearest degree and all ratios to the nearest ten thousandth. 1) tan Z = 0.1944 14 5 3) cos Q m<Z = __________ 2) Cos (10) = x x = __________ m<Q = __________ 4) Sin(40) = x x = __________ Round all side measures to the nearest tenth and all anges to the nearest degree 5) 6) x x = __________ 25 x = __________ x 20 7) 8 5 x 75 32 8) 10.8 9) 10) 5 3 x x = __________ 38 x = __________ 51 x = __________ 6 x = __________ x x 11) 12) 400 x 81 x 25 x = __________ 20 x = __________ 13) x 14) 24 6 x 4 26 27 28 29 30 9. 10. 11. 12. 14. 13. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 31 30. 31. 32. 7.5 word problems 1) 2) 3) 4) 5) 6) 7) 8) 32 7.5 book problems 1. 2. 3. 4. **1 kilometer = 1000 meters 33 5. 6. 7. 8. 34