Homework Assignments Math /02 Fall 2014

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1 Homework Assignments Math /02 Fall 2014 Assignment 1 Due date : Friday, September 5 6th Edition Problem Set Section 6.1, Page 178: #1, 2, 3, 4, 5, 6. Section 6.2, Page 185: #1, 2, 3, 5, 6, 8, 10-14, 16, 17, 18, 20, 22, 27. Read and study Euclid s constructions of: an equilateral triangle, the bisection of an angle, the bisection of a straight line segment, and the construction of a perpendicular thru a given point. They all may be found in the article Then solve the following two construction problems in the style of Euclid. 1. At a given point on a given line describe a compass/straight edge construction of an angle equal to a given angle. Also, provide a proof that your construction is valid by using properties of congruent triangles. 2. On a given line segment describe a compass/straight edge construction of a perpendicular bisector. Then provide a proof that your construction is valid by using properties of congruent triangles. Assignment 2 Due date : Monday, September 15 6th Edition Problem Set Section 6.3, Page 189: # 1(a)(c), 2(a)(c)(f), 4, 7, 11, 12, Carefully prove that the line joining the midpoint of any chord to the center of the circle is the perpendicular bisector of that chord. Support each step in your solution. You should include a figure and invoke an appropriate Euclidean criterion for congruent triangles. 2. Write step-by-step instructions on how to draw a circle using only compass and straight edge thru three non-collinear points A, B and C in the plane. 3. A regular n-sided polygon, A1A2... An, has each of its sides of equal length and each of its interior angles of equal measure. It has been mentioned in class that any regular polygon, A1A2... An, can be circumscribed by a circle. This fact follows immediately from the following three propositions which you must now carefully prove: Proposition 1: If O is the point where the angle bisectors of A n A 1 A 2 and A 1 A 2 A 3 intersect then A1OA2 is isosceles. Use properties of congruent triangles here to prove that a triangle with two equal angles is isosceles. Proposition 2: If O is the point described in Proposition 1 then A 1 OA 2 y A 2 OA 3 y... y A n OA 1. Be careful when proving this, you cannot just assume that A3O is the angle bisector of A 2 A 3 A 4. Proposition 3: A 1 O = A 2 O =... = A n O. Now the circle centered at O with radius A 1 O passes thru all n vertices of the polygon. Since a circle is convex, each of its chords must lie wholly within its interior. Thus, this circle passing thru all of the vertices of A 1, A 2,..., A n will necessarily circumscribe it.

2 Assignment 3 Due date : Wednesday, September 24 6th Edition Problem Set Section 6.4, Page 193: #3, 7, 8, 9, 10, 11, 12, 16, 19, 22. Section 6.5, Page 199: #1, 3, 9(a), 12, 13, 14, 27, 29, 32, Describe how to drop an altitude (i.e. a perpendicular segment) from the vertex of a triangle to its opposite side using only straight edge and compass. Then justify your construction based on properties of congruent triangles. 2. The geometric mean or mean proportion b between the numbers a and c is the square root of the product of a and c. That is, b = ac. To find the geometric mean between AB and BC, Euclid marked off the lengths AB and BC, end to end, along the same straight line. Then he drew a semicircle on AC, and erected a perpendicular BD at B. He claimed that BD is the mean proportional between AB and BC. Prove that Euclid s construction is correct. 3. The golden (or extreme) ratio, Φ, was defined in The Elements by Euclid to be that ratio which divides a whole so that: the whole is to the greater as the greater is to the less. Thus, 1 + Φ : Φ = Φ : 1 or 1+Φ Φ = Φ 1 (a) Start with the above equation involving Φ, clear fractions, and then either complete the square or use the quadratic formula (see page 369 of the textbook) to prove that 1+ 5 Φ =. 2 (b) Verify that the construction described below yields the golden ratio: AG = Φ.. Start with the square ABCD, 1 unit on a side, and bisect it vertically with line EF. Then swing an arc of radius EC to the extension of line AED. Now the length of AG will be Φ. In problems 4 and 5 use Ptolemy s Theorem: If a quadrilateral is inscribed in a circle then the sum of the products of its two pairs of opposite sides is equal to the product of its diagonals. to prove the following theorems. 4. If equilateral triangle ABC is inscribed in a circle and Q is a point on the circle then the distance from Q to the most distant vertex of ABC is equal to the sum of the distances from Q to the two nearer vertices. So relative to the figure, you must prove that q = p + r. 5. (See next page.)

3 5. In any regular pentagon the ratio of the length of a diagonal to the length of a side is the golden ratio, Φ. Hint: Relative to the figure which displays the regular pentagon ABCDE, apply Ptolemy s Theorem to the cyclic b quadrilateral ABCD to prove that Φ = a = (For a proof of Ptolemy s Theorem, see ) Assignment 4 Due date : Friday, October 3 Section 6.6,Page 203 #2, 3, 14, 15, 23, 24(a). Section 7.1, Page 211: #1(a)(c), 2(a)(b), 3(a)(h)(p), 5, 8, 9, 10, 14. Section 7.2, Page 215: #1, 4, 7, 11, 15, 20. Section 7.3, Page 217: #2, 10, 16, 17, 20, 22, 23, 28, 29, 37. Extra-Credit Exercise 1. Construct the Great Pyramid Star shown in the figure on a separate sheet using a compass and straightedge by carrying out the following steps: i. Draw a horizontal segment with left endpoint O. ii. Use a construction method described in Assignment 2 to locate points A and B on the segment so that OA = 1 and AB = Φ, (the golden ratio). iii. Draw the circles of radii OA and OB with center O. iv. Circumscribe a square around the circle of radius OA as shown. v. Draw four congruent outward pointing isosceles triangles, each having its base along a distinct side of the square and a vertex point on the outer circle. We call a triangle having the dimensions of ABC an Egyptian triangle. It is the case that the dimensions of BCD are proportional to those of a face of the Great Pyramid. So upon folding the four triangles upward and joining them along their adjacent edges, one would obtain a pyramid proportional to the Great Pyramid of Giza. Assignment 5 Due date : Friday, October 10 Section 8.1, Page 236: #3, 4, 8, 15, 22, 23, 24, 28, 29. Section 8.2, Page 239: #1, 2, 5, 7, 9, 14, 15, 16, 20, 24, 27, 28, 29, 34. Section 8.3, Page 245: #1(a), 7(c), 13. Section 8.4, Page 250: #1(a), 7(b), 13. Exam 1 Review Topics Exam 1 Friday, October 10, Life Sciences Auditorium - Room 111, 8:35am 11:15am.

Assignment 6 Due date : Wednesday, October 29 Required Exercises 1 & 2. As discussed in the article, http://mypages.iit.edu/~maslanka/solargeo.pdf, S. R. Crown Hall may be represented as a rectangular box having the dimensions: 220 ft x 120 ft x 24 ft.

4 Assignment 6 Due date : Wednesday, October 29 Required Exercises 1 & 2. As discussed in the article, S. R. Crown Hall may be represented as a rectangular box having the dimensions: 220 ft x 120 ft x 24 ft. The figure below shows an overhead sketch of this structure on the day of an equinox at 4 hours after solar noon. It is noted on page 13 of the SolarGeo article that its shadow ( i.e., the green polygonal region ) extends Δ x = 55.8 ft eastward and Δ y = 21.5 ft northward of the structure at this instant. So if O = ( 0, 0 ), P = ( 220, 0 ), Q = ( 220, -120 ) and R = (0, -120 ) denote the coordinates of the points at the base of the building then the vertices of the shadow at this time are at the points: S = (55.8, 21.5 ), T = ( , 21.5 ) and U = ( , ). 1. Use the equations (5), (7), and (9) in the SolarGeo article to determine the angles ω, α, and Α on the day of the winter solstice at IIT at each of the following times: (i) 2 hours before local solar noon. (ii) 3 hours after local noon. Use these angles to determine the shadow length and the eastward/westward and the northward extent of the shadow cast by Crown Hall at each time: (i) and (ii). Then draw a top view sketch displaying Crown Hall and the vertices of its shadow at each time: (i) and (ii). 2. Use the distance formula along the surface of the Earth ( refer to formula (5) for SDPR, in the table: ) to find the surface distance between Chicago, IL (Latitude = 41.8 o, Longitude = 87.3 o ) and each of the cities: (a) Beijing, China: Latitude = 39.9, Longitude = (b) Sydney, Australia: Latitude = 33.9, Longitude = (c) Buenos Aires, Argentina: Latitude = 34.6, Longitude = 58.4 Assignment 7 Due date : Wednesday, November 5 Section 14.1, Page 411: #1, 5, 7, 14, 19, 22, 29, 34, 36, 41, 47, 53, 57, 73, 78, 79, #83 = bonus problem. Section 14.2, Page 415: #1, 3, 8, 11, 15, 16, 17, 18, 21, 22, 25, 27, 28, 33, 34. Section 14.3, Page 419: #4, 11, 12, 16, 18, 19, 20. Assignment 8 Due date : Wednesday, November 12 Section 16.2, Page 473 #1, 2, 3, 5, 9, 10, 11, 12, 13, 14, 15. Exercises #1 3 (below). Instructions In problems 1-3 plot at least once cycle of each of the sine or cosine waves. Plot all of the curves for each problem on a single graph. Do not use a calculator but rather follow the method outlined in section 15-2 of the 6 th ed. on pages

5 In each problem, you should obtain the graph of equations (b) and (c) from that of equation (a) by using a suitable horizontal or vertical shift of the previous graph in each instance. Be certain to: ( i ) Locate the coordinates of all the zeros ( I.e. the points of the form ( x, 0 ) ) on your plots. ) ( ii ) Locate the coordinates of all the extreme points ( I.e. the points of the form ( x, d + a ) and ( x, d a ) where: a = amplitude of the graph and d = average y height of graph. ) (iii) Locate the coordinates of all the middle points of the form: ( x, d + a / 2 ) and ( x, d a / 2 ). 1. ( a ) y = 2 sin( 3x ) ( b ) y = 2 sin ( 3x π ) ( c ) y = 2 sin ( 3x π ) 4 2. ( a ) y = cos ( x ) ( b ) y = cos ( x +π / 3 ) ( c ) y = cos ( x + π / 3 ) ( a ) y = 4 sin( 2x ) ( b ) y = 4 sin( 2 x π / 2 ) Extra-Credit Drawing Project Due date : Monday, December 1. Follow the instructions in the article: and reproduce each of its nineteen drawings by using only compass and straight-edge. Display each drawing on a separate 8.5 x 11 inch sheet. Assignment 9 Due date : Wednesday, November 19 Exercises #1 14 (below). Relative to each equation in 1 12, do the following: (a) Graph the equation in the rectangular θ r - plane over the specified θ range. (b) Graph the equation in polar coordinates in the xy - plane (where x = r cos θ and y = r sin θ). (c) Convert the equation for the polar plot from (b) to xy - coordinates. 1. r = 6, 0 < θ < 2π 2. θ = π π π, < θ < r = cos θ, 0 < θ <2π 4. r = cos θ, 0 < θ < 2π 5. r = cos θ, 0 < θ < 2π 6. r = sin θ, 0 < θ < 2π 7. r = 2 sin 2θ, 0 < θ < 2π 8. r = 3 sin 3θ, 0 < θ < π 9. r = 3 sin θ, 0 < θ < π 10. r = 3, 0 < θ < π 11. r 2 π = 9 sin 2θ, 0 < θ < sin θ r 2 = 16 cos 2θ, π π < θ < A slice of pizza occupies one twelfth of the area of a circular disc of radius 7 inches. If the slice is in the first quadrant, with one edge along the positive x-axis, and the center of the pizza at the origin, then specify appropriate inequalities for r and θ which describe this slice in polar coordinates. 14. Prove that the circle with center ( a, b ) and radius R : ( x a ) 2 + ( y b ) 2 = R 2 has the corresponding polar equation: r 2 2 r ( a cos θ + b sin θ ) = R 2 a 2 b 2. Exam 2 Review Topics Exam 2 Friday, November 21, Life Sciences Auditorium - Room 111, 8:35 11:15 am

6 Assignment 10 Due date : Wednesday, December 3 Section 22.1, Page 686 #40, 43, 51, 52. Section 22.2, Page 691 #3, 7, 10, 24, 25, 27, 28. Section 22.3, Page 700 #4, 7, 10, 19, 23. Exercises #1 9 (below). 1. Find the slopes of the lines through the two pairs of points; then determine whether the lines are parallel, coincident, perpendicular or none of these. ( a ) ( 1, 2 ), ( 2, 11); ( 2, 8 ), ( 0, 2 ) ( b ) ( 1, 5 ), ( 2, 7); ( 7, 1 ), ( 3, 0 ) ( c ) ( 1, 1 ), ( 4, 1 ); ( 2, 3 ), ( 7, 3 ) ( d ) ( 1, 1 ), ( 5, 2 ); ( 9, 5 ), ( 3, 4 ). 2. Find the slope of the line bisecting the angle from l1 to l2 with slopes m1 and m2 respectively. ( a ) m1 = 3 and m2 = 5 ( b ) m1 = 4 3 and m2 does not exist. Problems 3 7 refer to the triangle in the xy - plane having the vertices: A = ( 1, 5 ), B = ( 3, 1 ), and C = ( 1, 1). 3. Sketch a graph of ABC in the coordinate plane. 4. Find the xy -equations of all the three sides of ABC. 5. Use slopes to find the measures of each of the three interior angles in ABC. 6. A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side. Find the equations of the three medians of ABC. x Hint: Recall that the midpoint of segment PQ has coordinates ( P + x Q y, P + y Q ). 7. Sketch the medians of ABC in your graph from problem 3 and verify algebraically that all three medians intersect at the common point: S = ( x A + xb + xc, y A + yb + y C ). 3 3 Point S is known as the centroid or barycenter of ABC. 8. Find the equation of the perpendicular bisector of the segment joining ( 4, 2 ) and ( 2, 6 ). 9. (a) Find the equations of the lines that form the square in the figure. (b) Find the equations for the boundaries of the three circular discs in the figure. Hint: First find the radii of each circle. One easy way of doing this, is to use the formula for the radius of an incircle. See page 4 of the article for details. -1 y 1 1 x -1 Assignment 11 Due date : At Final Exam (Dec 8 or 11) Section 22.4, Page 707 #1, 2, 6*, 7*, 14, 15, 17. *Note: The vertex is at the origin in these problems. Exercises #1 5 (below).

7 1. Find the distance from the point to the given line by using the distance theorem ( a ) 2 x 4 y + 2 = 0 ; ( 1, 3 ) ( b ) 4 x + 5 y 3 = 0 ; ( 2, 4 ) 2. Find the distance between the parallel lines: L1 : 2 x 5 y + 3 = 0 ; L2 : 2 x 5 y + 7 = o. (Hint: It suffices to compute the distance from any particular point P1 on L1 to L2.) 3. Find the equation of the line L bisecting the angle from L1 to L2 given L1: 3 x 4 y 2 = 0 ; L2: 4 x 3 y + 4 = 0. (Hint: Note that if P = ( x, y ) is any point on the bisector L then its coordinates satisfy the condition: A 1x + B1 y + C1 A 2x + B2 y + C 2 = (*) A 1 + B1 A 2 + B2 where L1 : A 1 x + B1y + C = 0 and L2 : A x B y C =. The solutions to (*) yield the equations of both lines bisecting the angles between L1 and L2.) 4. Find the two points of intersection of the circles: x 2 + y x + y 26 = 0 ; x 2 + y x y 15 = Find the equation of the parabola P with focus F = ( 1, 1 ) and the directrix D: x + y = 0 in two ways: ( a ) By using the fact that : Q = ( x, y ) on P d ( Q, F ) = d ( Q, D ). ( b ) By rotating the parabola y 2 = x by 45 o about its axis and translating 1 1 its graph + units horizontally and + units vertically. Hint: In part (b), transform the equation of the parabola to polar coordinates before rotating this curve. Then transfer back to xy coordinates in order to translate the equation of the rotated parabola. Refer to the handout: for details on rotation and translations theorems. (Final) Exam 3 Review Topics Final Exam: Math Monday, December 8, 8:00-10:00am, Hermann Hall Math Thursday, December 11, 8:00-10:00am, 129 LS

Homework Assignments Math /02 Fall 2017

Homework Assignments Math /02 Fall 2017 Homework Assignments Math 119-01/02 Fall 2017 Assignment 1 Due date : Wednesday, August 30 Section 6.1, Page 178: #1, 2, 3, 4, 5, 6. Section 6.2, Page 185: #1, 2, 3, 5, 6, 8, 10-14, 16, 17, 18, 20, 22,