Math 105 Final Exam Practice Name:____________________ 1. a) Give the formula for the sum of interior angles of any regular polygon:________________ b) Give the formula for one interior angle of any regular polygon: _____________________ 2. Answer the following questions about a regular octagon: a What is the sum of the interior angles:_______________ b What is one interior angle:_____________ c What is one exterior angle:_____________ d What is the sum of the exterior angles:_____________ e Will an octagon tessellate? _____________ 3. One interior angle of an unknown regular polygon is 157.5, answer the following questions: a Give an equation that you could use to find how many sides this regular polygon has: Equation:___________________________________ b Solve your equation in the following space and give your final answer: Answer:____________________ 4. If the hypotenuse of a triangle is 20 cm and one of the legs is 12 cm. Use the space below to solve for the missing leg: 5. a) Give the distance formula: ______________________ b) Find the distance between the two points (-3,4) and (5,-1). If the answer is not a whole number, round to the 2 place values behind the decimal. 6. Explain what this picture is describing: _______________________________________________________ _______________________________________________________ _______________________________________________________ _______________________________________________________ _______________________________________________________ _______________________________________________________ 7. Draw all lines of reflection for each of the following: a) b) c) d) 8. Which letters in #7 do not have rotational symmetry? _______________ 9. What is the degrees of rotational symmetry for d) of #7? ______________ 10. Take the point (-6,3) and translate it 8 units to the right and 7 units down, what are the new coordinates? Answer: ( , ) 11. Take the point (-6,3) and reflect it over the x-axis, what are the new coordinates? Answer: ( , ) 12. Take the point (-6,3) and reflect it over the y-axis, what are the new coordinates? Answer: ( , ) 13. Take the point (-6,3) and rotate it 180 degrees around the origin, what are the new coordinates? Answer: ( , ) 14. Use the following 3D points and vectors to answer the questions below: A(3,4,5) B(-1,6,2) m<2,5,3> n<4,-7,1> a) Find the distance from the origin to point A. b) Reflect point B over the xz plane, what are the coordinates for B': _______________ c) Find the length of vector m. d) What is the resulting vector of 2n + m e) Draw vector m below: 15. Use the following matrices to answer the questions below: 1 3 A 2 5 3 2 B 1 4 6 2 C 2 3 4 5 7 1 D 3 4 0 2 a. Give the dimensions of D:______________ b. Calculate A - B c. Calculate C + D d. Calculate -3B Solution:___________ Solution:___________ Solution:___________ e. Calculate BA f. Calculate CB g. Calculate CD Solution:___________ Solution:___________ Solution:___________ 16. Give the dimensions of two matrices that you could not add together:________________ 17. Give the dimensions of two matrices that you could not multiply:_______________ 18. Give the dimensions of two matrices with different dimensions that you can still multiply together:______________________ 19. True or False If you can multiply two matrices together, then it must be possible to multiply them in reverse order. 20. True or False When you multiply a 3x2 to a 2x5, it would make a 3x5 matrix. 21. True or False The order doesn't matter in matrix multiplication (AB is the same as BA) 22. a) Plot the matrix in the graph below and label the points: A 3 2 B 4 5 C 6 4 b) Rotate the triangle 180 degrees around the origin. Draw and label the new triangle A'B'C' c) Place the coordinates of the rotated triangle into the blank matrix below: A' B ' C ' d) What matrix would you multiply to the original ABC matrix to get A'B'C': _______________ 23. What matrix causes a reflection over the x axis (using multiplication): _____________ 24. Multiply the original triangle from #22 to a) Give the answer: A' B ' C ' 0 1 1 0 b) What transformation occurred:_____________________ 25. Explain why parallel lines do not exist in spherical geometry: ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ 26. Draw all possible points that are the same distance away from (5,1) and (-3,4) in taxicab geometry: 27. A pizzeria decides to deliver pizzas on bike and they will not deliver over 5 blocks. a) Draw any point to represent the shop and show where they won't cross for deliveries. b) What is the name of the shape that this border makes