Pythagorean Theorem: 9. x 2 2
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1 Geometry Chapter 8 - Right Triangles.7 Notes on Right s Given: any 3 sides of a Prove: the is acute, obtuse, or right (hint: use the converse of Pythagorean Theorem) If the (longest side) 2 > (side) 2 + (side) 2 then the is an If the (longest side) 2 < (side) 2 + (side) 2 then the is an. If the (longest side) 2 = (side) 2 + (side) 2 then the is a. Pythagorean Theorem: Pythagorean Triple: For problems 13.--, determine if it is possible to form a triangle with the given sides. If so, tell whether the triangle is right, acute, or obtuse , 8, ,, ,, , 7 2, , 8, , 8, , 9, , 2, , 4, 10.,, , 8, , 11, , 40, 41 E. Pythagorean Theorem - Solve for. (Simplest radical form) = = = = = = = = PYTHAGOREAN THEOREM Show all necessary work. Make a sketch if needed. 1. A rectangle has length 2.4 and width 1.8. Find the length of a diagonal. 2. A rectangle has a diagonal of 2 and length of 3. Find its width. 3. Find the length of a diagonal of a square with perimeter 16 m. 4. Find the length of a side of a square with a diagonal of length 12 inches.. The diagonals of a rhombus have lengths 16 and 30 cm. Find the perimeter of the rhombus. 6. The perimeter of a rhombus is 40 cm, and one diagonal is 12 cm long. How long is the other diagonal? 7. Find the third side of a right triangle if the hypotenuse is 14 km and one side is 9 km. 8. A rectangle is 6 ft long and 11 ft wide. What is the length of the diagonal of the rectangle? 9. A pole is 10 ft high. A wire is attached to the top of the pole and fastened to an anchor in the ground. The anchor is ft from the bottom of the pole. What is the length of the wire? 10. A tower casts a shadow 40 m long. The distance from the top of the tower to the end of the shadow is 0 m. How high is the tower? 11. Marie and Kevin hiked 3 miles east and then 6 miles north. How far were they from their starting point? E. Find the value of
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3 .8 Special Right s Find the value of the variables. Use the given length to find each of the remaining two lengths. 1. g = 6 2. h = f = 9 4. f = 7. g = h = 7. t = 4 8. u = u = s = t = 12. s = 8 h 4 f g 3
4 Use the diagram to find the remaining lengths. 13. a = 2, b =, c =, d=, e = 14. a =, b =, c =, d=, e = 4 1. a =, b =, c = 10, d=, e = Use special right s to solve: 1. Find the length of a diagonal of a square with sides 10 inches. 2. Find the perimeter of a square whose diagonal is 4 cm. 3. One side of an equilateral has length 6 cm. Find the length of the altitude. 4. Find the perimeter of an equilateral if the altitude has length 9 cm.. Find the length of a side of an equiangular whose altitude is 12. 4
5 Trigonometric Ratio Abbreviation Definition Sine of P Sin P opposite leg to P hypotenuse O H Cosine of P Cos P adjacent leg to hypotenuse P A H Tangent of P Tan P opposite adjacent leg leg to P to P O A E 1 : Given the following right triangle, write the trig ratios in fraction and decimal form. Sin P = Sin Q = Cos P = Cos Q = Tan P = Tan Q = E 2: Find the following trig values using the calculator. (Make sure you calculator is in degree mode and then round to four decimal places) Sin 38 = 0.61 Sin 2 = Cos 38 = 7 Cos 2 = Tan 38 = Tan 2 = In Eamples 3-11, find the value of the missing variable(s).
6 6
7 Angle of Elevation: Angle of Depreession: 7
8 8.4 Make a sketch with a right, label, show trig equation. 1. A tree casts a shadow 21m long. The angle of elevation of the sun is 1. What is the height of the tree? 2. A helicopter is hovering over a landing pad 100 m from where you are standing. The helicopter s angle of elevation with the ground is 12. What is the altitude of the helicopter? 3. You are flying a kite and have let out 80 m of string. The kite s angle of elevation with the ground is 40. If the string is stretched straight, how high is the kite above the ground? 4. A 1 m pole is leaning against a wall. The foot of the pole is 10 m from the wall. Find the angle the pole makes with the ground.. A guy wire reaches from the top of a 120m television transmitter tower to the ground. The wire makes a 63 angle with the ground. Find the length of the guy wire. 6. An airplane climbs at an angle of 18 with the ground. Find the ground distance the plane travels as it moves 200 m through the air. 7. A lighthouse operator at a point 2 m above sea level sights a sailboat. The angle of depression of the sighting is 10. How far is the boat from the base of the lighthouse? 8
9 8. A 20 ft ladder leans against a wall so that the base of the ladder is 8 ft from the base of the building. What angle does the ladder make with the ground? 9. At a point on the ground 0 ft from the foot of a tree, the angle of elevation to the top of the tree is 3. Find the height of the tree. 10. From the top of a lighthouse 210 ft high, the angle of depression of a boat is 27. Find the distance from the boat to the foot of the lighthouse. The lighthouse was built at sea level. 8. The Law of Sines and Law of Cosines Eample 1 - Solve the triangle. Round to the nearest tenth. a. b. c. Use the Law of Sines to find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. 9
10 BC DE GH m J m R m T Use the Law of Cosines to find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree YZ BD EF m I m M m S Review Simplify Find the geometric mean between the given numbers in simplest radical form.. 2 and and and 2 I. State whether a with sides of the given lengths is acute, right, or obtuse , 8, , 9, , 13, , 6, A right has a 60 angle. If the length of the longer leg is 48, find the lengths of the shorter leg and the hypotenuse. II. Find the values of the variable in simplest radical form y
11 y In CDE, m C = 90, m D = 30. Which side of the is the longer leg? A tower casts a shadow 3 m long when the angle of elevation of the sun is 32. How high is the tower to the nearest tenth? 28. A rectangle has a diagonal of 40 meters and a width of 24 meters. Find the length of the rectangle. 29. An equilateral triangle has a perimeter of 30 inches. Find the length of its altitude. For Eercises 12 and 13, substitute numbers into the given Law of Sines ratio to find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree sin D sin C CE DE DE sin Q sin R PR PQ m R Use the Law of Sines to find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree EF m N For Eercises 16 and 17, substitute numbers into the Law of Cosines to find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. 16. TU 17. m H TU 2 ST 2 SU 2 2(ST)(SU)(cos S) GI 2 GH 2 HI 2 2(GH)(HI)(cos H) 11
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