Geometry Rules! Chapter 8 Notes

Transcription

1 Geometr Rules! Chapter 8 Notes Notes #6: The Pthagorean Theorem (Sections 8.2, 8.3) A. The Pthagorean Theorem Right Triangles: Triangles with right angle Hpotenuse: the side across from the angle (the side of the triangle) Legs: the sides across from the angles (the two sides of the triangle) Pthagorean Theorem (leg) 2 + (leg) 2 = (hpotenuse) 2 Solve for : Find and label c across from the right angle Label a and b Write and solve a 2 + b 2 = c 2 1.) 2.) ) ) 2 2 3

2 - 2-5.) ) A rectangle has length of 4cm and a width of 2cm. Find the length of its diagonal. (Hint: draw a picture first!!) 24 7.) The perimeter of a square is 20in. Find the length of its diagonal. 8.) The diagonals of a rhombus have lengths of 6ft and 8ft. Find the perimeter of the rhombus. (Hint: what do we know about the diagonals of a rhombus?) B. Classifing Triangles We can also use the Pthagorean Theorem to classif a triangle Acute: Right: Obtuse: a + b > c a + b = c a + b < c 2 2 2

3 - 3 - Classif the triangle with the given sides as acute, right, or obtuse. If the triangle is not possible, sa so: Check that the triangle is possible (short side + short side > long) Compare a 2 + b 2 vs. c 2 9.) 6, 8, ) 5, 6, 7 11.) 2, 4, 6 12.) 2, 5, 6 C. Algebra Practice: Solving Quadratics Get all terms to one side and equal to zero. Arrange in descending order Factor completel Set each ( ) = 0; solve each equation Solve 13.) = 0 14.) 2 3 = ) 2 = ) = 2 17.) = 3 + 5

4 Notes #7: Geometric Means and Similar Right Triangles (Section 8.1) A. Geometric Mean asks the question: what number, squared, equals the product of two given numbers? Find the geometric mean of the listed numbers: Use the given numbers in this equation: 2 = ab Solve for 1.) 9 and 16 2.) 12 and 3 3.) 5 and 15 B. Similar Right Triangles When an altitude of a right triangle is drawn to its hpotenuse, three similar right triangles are formed: z a b

5 - 5 - Solve for the variables: Re-draw the three triangles and label all sides Set up proportions to solve for the variables Look for was to use the Pthagorean theorem 4.) m n p ) a b c

6 - 6-6.) 3 5 z C. Algebra Practice Solve for b factoring: 7.) = ) = 16 9.) 3 3 = ) 2(2 2 5) =

7 - 7 - Notes #8: Special Right Triangles (Section 8.4) A. 90 Triangles Solve for the missing sides using ITT and Pthagorean Theorem Write a rule based on the pattern 1.) 2.) 3.) Triangles Solve for : Find the side for which ou have a value Set this side = to its rule Solve for n Plug n back into the triangle rules find the length of all sides 4.) 5.) 6 2 3

8 - 8-6.) 7.) ) 9.) A. 90 Triangles Look for the pattern in the triangles below a) b) Triangles

9 Solve for the missing sides: Find the side for which ou have a value Set this side = to its rule Solve for n Plug n back into the triangle rules to find the length of all sides 10.) 11.) ) 13.) ) 15.) ) 17.) 8 10

10 Notes #9: Right Triangle Trigonometr (Sections 8.4, 8.5, 8.6) Solve for the variables: 1.) 2.) ) 4.) ) 6) 40 z 3 6 z 2

11 B. 3 Trig Functions Trigonometr relates a right triangle s to its. Sine(angle) = (angle) = Cosine(angle) = (angle) = Tangent(angle) = (angle) = OR (angle) = Complete the triangle and find each value as a simplified fraction: 7.) sina 8.) sinb B 9.) cosb 10.) cosa 10 4 A C 11.) tana 12.) tanb C. Using our Trig Table If ou know the angle, find the angle in the left-most column and read to the right to find its sine, cosine, and/or tangent If ou know the decimal value of its sine, cosine, and/or tangent, look down the sin/cos/tan column until ou find the closest match. The read to the left to find the angle. Use our trig table. Round our answers to the nearest hundredth: 13.) a) sin32 = b) tan19 = c) cos75 = d) cos48 = e) sin80 = f) tan59 =

12 Use our trig table. Round our answers to the nearest degree: 14.) a) sin = b) tan = 0.21 c) cos = d) cos = 0.4 e) sin = 0.7 f) tan = 2.5 D. Using our Calculator (make sure our calculator is in degree mode!!) If ou know the angle, tpe the sin/cos/tan button, then the angle, then enter/equals. It should look like this on our calculator screen: sin(37) = If ou know the decimal value of its sin/cos/tan, tpe the 2 nd /Shift Ke, then the decimal, then enter/equals. It should look like this on our calculator screen: cos -1 (0.5) = Use our calculator. Round our answers to the nearest hundredth: 15.) a) sin32 = b) tan19 = c) cos75 = d) cos48 = e) sin80 = f) tan59 = Use our calculator. Round our answers to the nearest degree: 16.) a) sin = b) tan = 0.21 c) cos = 0.79 d) cos = 0.4 e) sin = 0.7 f) tan = 2.5 E. Solving Quadratics using the Quadratic Formula Not all quadratics (a 2 + b + c = 0) can be factored 2 b b 4ac ± In this case, use the quadratic formula: = 2a Reduce the radical epression and reduce the fraction, if possible 17.) = 0 18.) = 0

13 19.) 2 3 = 1 20.) = Notes #10: Right Triangle Trigonometr (Sections 8.5, 8.6, 8.7) A. Constructing Right Triangles Complete without our calculator: 3 1.) If sin A =, find tan A. 2.) If 5 cos B = 2, find sinb. 3 3.) If 5 sin A =, find tan 2 A 8 4.) Find the altitude of an equilateral triangle with perimeter 12 in.

14 B. Solving for Missing Sides of Right Triangles Pick an acute angle; label sides as O(opposite), A(adjacent), and H(hpotenuse) Choose a trig function (sine, cosine, tangent), write an equation Solve for the variable wait to use our calculator until the last step Write an equation to solve for each variable. Round sides to the nearest tenth and angles to the nearest whole degree: 5.) z ) z

15 C. Solving for Missing Angles of Right Triangles Follow the same steps, but remember that when ou are solving for an angle, ou must use the 2 nd /Shift Ke (sin -1 ) and/or use our trig table from right to left. Find missing sides in reduced radical form and find missing angles to the nearest whole degree: 7.) 5 2 z 8.) z 10 6

16 Notes #11: Applications of Right Triangle Trigonometr A. Angles of Depression and Elevation (all relative to a horizontal line of sight) Solve each problem: Draw a picture, include: line of sight angle of depression/elevation labeled right triangle Write an equation and solve Solve for the missing information: 1.) The angle of elevation of a ramp is 25. If the ramp is 6m off the ground at its highest point, how long is the inclined surface of the ramp? 2.) A streetlight casts a 5ft shadow. If the streetlight is 9ft tall, what is the angle of elevation of the sun from the ground? 3.) Joe is out fling his kite. He has let out 100 ft of string and knows that the kite is 65ft off the ground. What is the angle of elevation of the kite string from the ground?

17 Classwork #11: Chapter 8 Review Special Right Triangles: Solve for and in reduced radical form. 1.) 2.) ) 4.) Similar Right Triangles: Solve for m, n, and p in reduced radical form. 5.) m n p 5 10 Word Problems: Leave answers in reduced radical form (no decimals!) 6.) The altitude of an equilateral triangle is 6ft. Find its perimeter. 7.) The perimeter of a square is 24m. Find the length of its diagonal.

18 ) The hpotenuse of a,, 90 triangle is 10in. What is the length of one of its legs? 9.) The diagonals of a rhombus are 10cm and 24cm long. What is the perimeter of the rhombus? Using Trigonometr: Solve for the indicated quantit. Round lengths to the nearest tenth and angles to the nearest whole degree. 10.) 11.) ) ) A flagpole casts a shadow that is ft long. If the angle of elevation to the sun is 31 degrees, how tall is the flagpole? 4 14.) A 12ft ladder leans against a building in such a wa that its base is 4ft from the building. What is the angle of elevation of the ladder to the building?

19 Notes #12: Circles and Ke Vocabular (Section 9.1) O B 1.) Circle O (written as O) has center 2.) OA is a of the circle. This is a segment connecting the to an on the circle. Other radii:, A C 3.) AB is a of the circle. This is a segment connecting two on the circle and passing through the circle s. 4.) What is the relationship between a radius and a diameter? 5.) AB and MN are. These segments connect an points on a circle. 6.) What is a name for the longest chord in a circle? M N A 7.) AB and MN are. These are lines that contain a. B 8.) A is a segment, ra, or line that touches a circle onl once. Name four tangents:,, O 9.) The point where a tangent touches a circle is called the. Name the point of tangenc: X Y Z 10.) A tangent is alwas to the radius at the point of tangenc. 11.) Name two right angles:,

20 ) Circles and spheres are called if the have the same center. 13.) Draw two concentric circles and two concentric spheres concentric circles concentric spheres 14.) We sa that a polgon is a circle when all vertices (corners) are on the circle. In this case, we can also sa that the circle is about the polgon. 15.) Describe this figure in two was: A B P C D 16.) Describe this figure in two was: G Q F H 17.) Draw a triangle inscribed 18.) Draw a circle circumscribed about in a circle. a rectangle.

21 Geometr Chapter 8 Stud Guide: Right Triangle Geometr Radical Epressions: Simplif each epression ( 3 5 ) 2 4. ( 4 6)( 3 2 ) Pthagorean Theorem: Solve for Find the length of a diagonal of a square with perimeter 20m. 9. The diagonals of a rhombus have length 8cm and 6cm. Find the perimeter of the rhombus. State whether the triangle with the given sides is not possible, right, acute, or obtuse: 10. 4, 6, , 4, , 10, 12 Geometric Means and Similar Triangles: 13. Find the geometric mean of 5 and Find the geometric mean of 4 and Solve for,, and z: (hint: use 3 triangles) 16. Solve for,, and z: (hint: use 3 triangles) z 4 z

22 Special Right Triangles: Solve for and Right Triangle Trigonometr: 26. Find sina, cosa, and tana as fractions C 27. Find sinb, cosb, and tanb as fractions B A 10 B C A Solve for and ; round to the nearest tenth or leave in radical form:

23 Applications of Trigonometr: 32. If the sun s angle of elevation is 48 and a flag pole casts a shadow that is 40 feet long, how tall is the flag pole? 33. From a lighthouse that is 150m above the shore, the angle of depression to a ship is 20. How far is the ship from the shore? 34. CCA students want to design a water park and have a particular slide in mind. The want it to be perfectl straight and 40ft high, and for students to have a ft long slide-ride. What must the angle of elevation of the slide be? (Round to the nearest whole degree.) 35. How far from the base of a building is the bottom of a ft ladder that makes an angle of 75 with the ground? Solve b Factoring: = = = = = = = = 4-12 Solve b using the quadratic formula: = = = ( + 5) = ( + 6) = = -6-2