Higher Geometry Problems

Higher Geometry Problems (1) Look up Eucidean Geometry on Wikipedia, and write down the English translation given of each of the first four postulates of Euclid. Rewrite each postulate as a clear statement in modern English, and illustrate with a picture. (2) Rewrite the English translation of Euclid s original fifth postulate using more clear, modern language. Draw a picture that illustrates the statement. () Write down Playfair s Postulate, and explain why it is equivalent to Euclid s fifth postulate. (4) Finish the definition: Two sets A and B in the plane are congruent if... (5) Explain why if we say that a line segment in the plane is vertical, this description of the line segment is not a geometric description. (6) If we consider a finite set S of points in the plane, explain why the number of points of S is a geometric quantity. (7) Give the precise definition of a triangle. (8) Give the precise definition of a quadrilateral. (9) Give the precise definition of a rectangle. (10) Give the precise definition of a parallelogram. (11) Give the precise definition of a circle in the plane. (Note: a circle does not include its insides.) (12) Give the precise definition of a disk in the plane. (1) Write down a precise proof that the sum of the measures of the (interior) angles in a triangle is π, the measure of a straight angle. (14) Write down a statement and proof of the formula for the area of a parallelogram. (15) Write down the precise definition of equality of angles. (16) Prove the exterior angle theorem. That is, let the triangle ABC be given. Let D be a point on the ray AB but not on AB. Then m CBD = m BAC + m BCA. (17) Prove that the fact that the sum of angles in a triangle is equal to a straight angle implies Playfair s Postulate. (Thus, the triangle fact may be used in place of the Parallel Postulate.) (18) Prove the formula for the area of a triangle. (19) Give the precise definition of an isosceles triangle and the precise definition of an equilateral triangle. (20) Prove that if triangle ABC is isosceles with AB = AC, then B = C. (21) Show how to construct the perpendicular bisector of a line segment (and thus its midpoint) using a collapsing compass and unmarked straightedge. Prove that your construction works. (22) Suppose that CD is the perpendicular bisector of the line segment AB. Prove that every point on CD is equidistant from A and B. (2) Prove the converse of the last statement. (24) Show how to construct a line that is perpendicular to a given circle at a given point, using a collapsing compass and unmarked straightedge. You may not assume that you have the center of the circle. Prove that your construction works. (25) Show how to construct the incenter of a given triangle, and prove that your construction works. 1

2 (26) Show how to construct a tangent line to a circle at a given point on the circle with an unmarked straight edge and collapsible compass. Prove that your construction works. (27) Explain how to construct the center of a given circle. Prove that your construction works. (28) Explain how to construct the diameter of a given circle that contains a given point on the circle. Prove that your construction works. (29) Prove the SAS triangle congruence theorem, using basic notions only. (0) Prove the ASA triangle congruence theorem. You may assume that the SAS triangle congruence theorem has been proved. (1) Suppose that l 1 and l 2 are two different lines that intersect at a point p in the plane. Let l be any other line in the plane. Prove that at least one of l 1 and l 2 must intersect l. [Make sure that your argument is short and concise.] (2) Prove that if L 1, L 2, and L are three lines in the plane, then if L 1 L 2 and L 2 L then L 1 L. () Prove the SAA triangle congruence theorem. You may assume that the ASA triangle congruence theorem has been proved. (4) Prove the SSS triangle congruence theorem. You may assume that the SAS triangle congruence theorem has been proved. (5) Prove that there is no SSA triangle congruence theorem. (Hint: give a counterexample.) (6) Given any line and a point not on the line, show how to construct the line that is perpendicular to the line and that goes through that given point, using a collapsing compass and unmarked straightedge. Prove that your construction works. (7) Given any line and a point not on the line, show how to construct the line that is parallel to the line and that goes through that given point, using a collapsing compass and unmarked straightedge. Prove that your construction works. (8) Prove that lengths can be copied, using a collapsing compass and unmarked straightedge. That is, assume that the line segment AB is given, and assume that another point C is given, and that a ray with vertex C is given. Prove that you can construct a point D on the ray so that AB = CD. (Hint: Draw AC. Construct lines perpendicular to AC at A and at C. Figure out a way to make new points E and F on these new lines so that AB = AE and then AE = CF. Use F to make the point D.) (9) By proving that lengths can be copied (as in the last problem) using a collapsing compass and unmarked straight edge, you are showing that you any construction made with a collapsing compass and unmarked straight edge can also be done with a rigid compass and unmarked straight edge. Therefore, to show that something can be constructed, it is sufficient to show that it can be constructed using a rigid compass and unmarked straight edge. This is the Rigid Compass Theorem. Explain the reasoning behind this. (40) Prove that angles can be copied using a collapsible compass and unmarked straight edge. You may assume that we have already shown the Rigid Compass Theorem. (That is, assume ABC is given and that the line segment DE is given. Prove that we may construct a point F such that ABC = DEF.) (41) Show how to bisect an angle using a straight edge and compass construction. Prove that your construction works.

(42) Show how to construct a 120 angle using an unmarked straight edge and collapsing compass. Give your answer as a sequence of steps. Diagrams are allowed, and they don t have to be pretty. Prove that your construction is correct. (4) Prove that a simple quadrilateral is a parallelogram if and only if its opposite sides are congruent. (44) Demonstrate that the word simple cannot be deleted from the previous exercise. (45) Let the simple quadrilateral ABCD satisfy AB = BC. Suppose that the diagonals AC and BD are perpendicular at point E on AC. Prove that CD = DA. (46) Prove that in a parallelogram, opposite angles are congruent. (47) Prove or disprove that the diagonals of a parallelogram bisect each other. (48) Prove or disprove that the diagonals of every simple quadrilateral bisect each other. (49) Prove or disprove that the diagonals of a parallelogram are congruent. (50) Prove or disprove that if the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. (51) Prove or disprove that if the diagonals of a quadrilateral are congruent, then the quadrilateral is an isosceles trapezoid. (52) Prove or disprove that if the diagonals of a trapezoid are congruent, then the quadrilateral is an isosceles trapezoid. (5) Define kite. (54) Prove or disprove that a quadrilateral is a kite if the diagonals are perpendicular. (55) Prove or disprove that if a quadrilateral is a kite, then the diagonals are perpendicular. (56) In the plane, you are given a line l and a point P not on the line. Show how to construct a line parallel to l through the point P, using an unmarked straight edge and collapsing compass. Give your answer as a sequence of steps. Diagrams are allowed. (You do not need to use a compass to make your diagrams pretty.) You do not need to justify the steps. (57) Prove that the number n is constructible, if n is any positive integer. (58) Prove that the length 2 is constructible. (You may assume that we know how to do the constructions we have done in class/notes.) (59) Prove that 75 is irrational. (60) Prove the following versions of the Carpenter s Rectangle Theorem: (a) Given a quadrilateral ABCD, suppose that B is a right angle and that AB = CD. Show that if the diagonals have the same length (i.e. AC = BD ), then ABCD is a rectangle. (b) Given a quadrilateral ABCD such that AB = CD and AD = BC, prove that if the diagonals have the same length (i.e. AC = BD ), then ABCD is a rectangle. (61) Prove that if ABCD satisfies AB = CD and ABC = BCD, then BAD = CDA. (62) Let P QR be an isosceles triangle, with P Q = P R. Let S and T be points of QR that are spaced so that QS = ST = T R = 1 QR. Construct the line segments P S and P T. Prove that P ST is an isosceles triangle. (6) Let ABC be a triangle such that the perpendicular bisector of AB intersects BC at its midpoint. Prove that ABC is a right triangle. (64) Prove that if an altitude within a triangle is the perpendicular bisector of the side it meets, then the triangle is an isosceles triangle.

4 (65) Let F GHJ be a simple quadrilateral. Prove that the sum of the measures of any three interior angles of the quadrilateral must be greater than 180 and less than 60. (66) Prove that 5 is irrational. (67) Give two different proofs of the Pythagorean Theorem. (68) Given any quadrilateral ABCD, prove that the midpoints of the sides are either collinear or form the vertices of a parallelogram. (69) Let ABC be any given triangle. Let M and P be points on the line segments AB and AC, respectively, such that AM = 4MB and AP = 4P C. Construct the line segment MP. Prove that ABC is similar to AMP. (70) True or False. Provide proof. (a) If a is a quadrilateral, then a is not a square. (b) If a is a rhombus, then a is a kite. (c) If a point X is on the perpendicular bisector of a line segment AB, then AX = BX. (d) Every rectangle has three sides. (e) All triangles are equilateral. (f) Some triangles are equilateral. (g) No isosceles triangle is a right triangle. (71) Find the converse of each statement above, and determine if it is true or false. Provide proof. (72) Suppose that ABC is given such that AB = 5, BC = 9, AC = 7. Let X, Y, Z be points on AB, BC, and AC, respectively, and suppose that AX = 2 and AZ = 4. If the line segments AY, BZ, and CX are concurrent, find BY. Justify your steps. (7) Let ABC be an equilateral triangle with side length 8. Derive the formula for the height of this triangle, using the Pythagorean theorem. Then find the area of the triangle. (74) An isosceles right triangle has side lengths x and x + 2. Find x. Justify the steps. (75) Let ABC be given, so that AB = 5, BC = 12, and AC = 1. Prove or disprove that ABC is necessarily a right triangle. (76) Let ABC be a triangle, and let AY, BZ, and CX be cevians. Suppose that the cevians are concurrent, and that AX = BY =, AZ = 2, CZ = 4, CY = 6. (a) Prove that ABC is an isosceles triangle. (b) Prove that Area( ACY ) = Area( BCZ). (c) Prove that ABC is not a right triangle. (d) Prove that if we construct the line segment Y Z, then ABC is similar to ZY C. (77) Let ABC be a triangle, and let AX, BY, and CZ be cevians. Suppose that the cevians are concurrent, and that BX = BZ = 1, CX = 2, AY = AZ =. Prove that ABC is a right triangle. (78) Suppose that a triangle ABC has points P on AB, Q on BC, and R on AC. Suppose that AP = BQ = x > 1 and BP = QC = CR = 1. Suppose in addition that the line segments AQ, BR and CP are concurrent. (a) Prove that AR = x 2. (b) Prove that ABC is not an equilateral triangle.

(79) Let ABC be a triangle, and let D be a point on side AB. If we construct CD, prove that Area ( ADC) Area ( BDC) = AD BD. (80) Let ABC be a given triangle, with points D on BC, E on AC, and F on AB. We construct the line segments AD, BE, and CF. Suppose that F B = = DB, and F A DC suppose that AD, BE, and CF are concurrent at a point p. Prove that E is the midpoint of AC. (81) Let XY Z be a triangle, let M be the midpoint of XY, and let N be the midpoint of Y Z. We construct the line segment NP, so that NP XY and so that P is on XZ. Also, construct MN (a) Draw a picture of this situation. (b) Prove that XMNP is a parallelogram. (c) If the area of XY Z is 40, compute the area of XMNP. (82) Let CD be a chord of a circle, i.e. a line segment such that C and D are on the circle. Prove that the center O of the circle is a point on the perpendicular bisector of CD. (8) Prove that if a parallelogram is a cyclic quadrilateral, then it is a rectangle. (84) Prove or disprove: If two angles of a cyclic quadrilateral are right angles, then the quadrilateral is a rectangle. (85) Let XY Z be an equilateral triangle, and let P, Q, and R be on XY, Y Z, and XZ, respectively. Suppose that XP = Y P = 1, Y Q = 1.5, and P, Q, and R are collinear. Find ZR. (86) State the converse to Ceva s Theorem. (87) Prove that, given three points of the plane that are not colinear, there exists a unique circle that contains the three points. (88) Suppose that an altitude of a given triangle divides the triangle into two similar triangles. Prove that either the original triangle is isosceles or that the original triangle is a right triangle. (89) Let DEF satisfy DE = 4, m D = 0, and m E = 100. Find the other sides and angles. (90) Let DEF satisfy EF = 4, DF =, DE = 6. Find the angles of DEF. (91) A triangle in the Euclidean plane has sides of length 0 and 40. If the angle between the two sides is 50, find all possible lengths of the third side. (92) Let ABC be a triangle such that AB = 7, BC = 8, AC = 10. Let D, E, F be points on AB, BC, AC, respectively, such that AD = 4, CE =. If CD, BF, and AE are concurrent, find CF. (9) Using analytic geometry, prove that if two sides of a simple quadrilateral are parallel and have the same length, then the quadrilateral is a parallelogram. Do not assume any other geometric theorems other than analytic geometry facts. (94) Let ABC be an acute triangle, and let X be the point on AC that is the foot of the altitude from the vertex B. Construct BX. Use analytic geometry to prove that the location of the orthocenter (on BX) is at height of h above AC, where h = (AX) (CX). (BX) 5

6 (95) Prove that if a triangle has an obtuse angle (i.e. an angle whose measure is > 90 ), then the orthocenter is not in the interior of the triangle. (96) We are given a triangle ABC and midpoints M, N, P of AB, BC, AC, respectively. Construct the triangle MNP. Prove using analytic geometry that Area( MNP ) = 1 Area( ABC). 4 (97) Using analytic geometry, prove that the medians of any given triangle ABC are concurrent (at the centroid G). Also, prove that the height of G above side BC is exactly 1 the height of A above BC. (98) Using analytic geometry, prove that the perpendicular bisectors of any triangle are concurrent. (99) Using analytic geometry, derive the formula for the distance between (a, b) and the line Ax + By + C = 0. (Hint: this distance is achieved by a perpendicular line segment.) (100) Using analytic geometry, prove that if in any triangle ABC, if D and E are the midpoints of AB and AC, respectively, then DE is parallel to BC. (101) Using analytic geometry, prove Thales Theorem: If A, B, and C are points on a circle where the line segment AC is a diameter of the circle, then the angle ABC is a right angle. (102) Give the degree measure of each angle. (a) π 5 (b) 1.24 (10) Give the radian measure of each angle. (a) 80 (b) 12.45 (104) Find the angle in radians and degrees corresponding to the location on the unit circle. (Note: give all possible angle measurements.) (105) For each angle θ given, find the (exact) (x, y) coordinates of the corresponding point on the unit circle, and find the values of cos (θ), sin (θ), tan (θ), sec (θ), csc (θ), cot (θ). If the trigonometric function is undefined, say so. (a) θ = 0. (b) θ = 5π 4. (c) θ = 11π 6.

(106) Complete the following trigonometry identities: (a) = sin (B) cos (C) + sin (C) cos (B) (b) sin ( ) 2 π 8 = (c) sin ( ) 2 π 8 = (different from last answer) (d) cot 2 (φ) + 1 = (107) True or False. Assume all variables are positive numbers and all expressions are defined. (a) cos 4 (θ) cos 2 (θ) = cos 2 (θ) (b) sin (A) csc (A) (sec (A)) 1 = cos (A) (108) Simplify and factor the result. Assume all variables are positive and that the expressions are defined. (a) cot (C) sec (C) sin (2C) (b) tan (B) cos (π + B) sin (B) (109) Solve the equation sin (θ) = 1 exactly, using the unit circle (not calculator). 2 (Note that all solutions θ should be given.) (110) Make a table of values using the unit circle. Then graph (no calculator). (a) f (θ) = sin 2 (θ) (b) g (θ) = cos ( θ 2) + 1 (111) Find an obtuse angle A such that sin (A) = 0.7. (112) Prove the Law of Cosines in the case when the triangle is acute. (11) Prove the Law of Sines in the case where the triangle is acute. (114) Find all possible angles θ (measured in radians) such that csc (θ) = 2. (115) Find all possible angles θ (measured in radians) such that cot (θ) =. (116) Solve the equations below. (a) sec (θ) = 2. (b) y = arcsec ( 2). (c) sin (A) = 1 2. (d) A = arcsin ( 1 2 ). (117) Find the values of all the trig functions at the angle 495. (118) Prove that (119) Prove that (120) Find (no calculator): (a) arcsin ( 1) (b) arctan ( 1) ( (c) arcsec cos (A) + cos (A) sin 2 (A) cos (A) cos (A) 2 = sin 2 (A). sin (B) tan (B) cos (B) = cos (2B) sec (B). ) (121) Define arctan (x). (122) Complete the following trigonometry identities: (a) cos 2 (A) + = 1 (b) cos (2θ) = (c) cos (2θ) = (d) cot 2 (φ) + 1 = (different from last answer) 7

8 (12) True or False. Assume all variables are positive numbers and all expressions are defined. (a) tan (y) = tan (y) (b) sin (x) = cos 2 x sin x sin x (124) Simplify and factor the result. Assume all variables are positive and that the expressions are defined. csc(c) sec(c) tan(c) sin(c) cos(c) cot(c) (a) (b) tan (B π) cos (π + B) sin (B) (125) Prove that for any θ R, sin (4θ) = 4 cos θ sin θ 4 cos θ sin θ. (126) By examining graphs find the smallest positive number B so that cos (θ) = sin (θ + B). (Assume θ is measured in radians.) (127) Prove that sin (A) = cos 2 A sin A sin A. tan (x) sin (x). (129) Simplify (csc (p) + 1) (csc (p) 1) (cos (2p) 1). (10) Prove that this statement is false: For any angle A, (128) Simplify sin(x) tan(2x) + 1 2 (sin (A)) = 1 2 sin A sin (A). 2 (11) Prove that tan (θ) tan (θ) = sin2 θ cos 2 θ sin 2 θ cos 2 θ (12) Suppose the lengths of the sides of a triangle in the plane are 4x, 5x, and 8x, for some positive real number x. Find the answers to these questions. (a) This triangle is (always//sometimes//never) a right triangle. (b) The angles of this triangle are (always//sometimes//never) all acute angles. (c) The area of this triangle is (always//sometimes//never) bigger than 4 units. (d) The perimeter of this triangle is (always//sometimes//never) equal than units. (1) A triangle has sides of length 12, 7, and 6. Using trigonometry and your calculator, find the angles: (a) angle opposite 6 side =? (b) angle opposite 7 side =? (c) angle opposite 12 side =? (14) Let ABC be the triangle with side lengths 5, 12, 1. (a) Find measurements of all the angles of the triangle. (b) If the median is constructed from a vertex to the side of length 1, find the measurements of the two angles made at that vertex. (15) A triangle has a side that is 4 units long and another side that is 8 units long, and the angle between the two sides is 22. Find: (a) Perimeter of the triangle (b) Area of the triangle (16) Suppose ABC has AB = 8, BC = 7, m A = π. Explain why there are two 5 possibilities for B, and find these two possible angle measurements. Also find the two possibilities for AC.

(17) Let DEF satisfy DE = 4, EF = 9, m DF E = 10. (a) Find all the other side lengths and angles. (b) Explain why there is more than one set of answers in (a). (18) A triangle in the Euclidean plane has two angles that measure 25 and 75, respectively. Suppose that the side opposite the smallest angle of the triangle has length 5.75. Find the perimeter and area of the triangle. (19) Let ABD be a triangle such that B is a right angle, and let C be the foot of the altitude from vertex B. (a) Prove that all three triangles present in the picture are similar. (b) Prove that AC CD = BC 2. (140) Write down a precise statement and proof of the Law of Sines. (141) Write down a precise statement and proof of the Law of Cosines. (142) Find the following, and draw a picture that demonstrates that your answer agrees with intuition. (a) ı + 5j (b) (, 2) + ( 5, 1) 2 (c) (, 2) ( 5, 1) 2 (d) ( ı + 2j) ( 2ı + j) 5 5 5 5 (e) (, ( 4 5 5) 4, ) 5 5 (f) ( ı + 2j) ( ı 2j) 5 5 5 5 (14) Find the vector equation of the line x 7y = 5. [The vector equation of the line is an equation of the form α (t) = U + tv, where U and V are two constant vectors, and where {α (t) : t R} is the set of all points on the line. U is often called the position vector, and V is often called the velocity vector.] (144) Find the set of all vectors v such that v (1, ) = 0. Draw a picture that supports your conclusion. (145) The triangle XY Z in R 2 is placed so that X = (0, 0), Y = (5, 0), XZ = 4, and Y Z = 2. Find all possible locations of vertex Z. (146) Find the set of all vectors w such that w (1, ) = 2. Draw a picture that supports your conclusion. (147) Perform ( the following ) ( matrix operations, ) if possible: 2 1 2 2 (a) + 7 0 2 ( ) ( ) 2 1 2 2 (b) 7 0 2 ( ) ( ) ( ) 2 1 2 2 (c) + 7 2 9 ( ) ( ) 2 1 x (d) 7 y ( ) ( ) 2 0 x (e) 0 7 y ( ) ( ) ( ) 0 1 x 2 (f) + 1 0 y 5 9

10 (g) ( 2 6 1 ) 1 2 9 4 2 1 1 1 (h) ( 2 6 1 ) 1 2 4 1 1 (i) ( 2 6 1 ) ( ) 1 2 9 1 1 1 (148) Prove( this particular ) example ( of the associative ) ( property ) of matrix multiplication. If A11 A A = 12 B11 B, B = 12 x1, x =, prove that (AB) x = A (Bx). A 21 A 22 B 21 B 22 x ( ) ( ) ( 2 2 1 2 4 ) ( ) ( ) (149) Let A =, B =, C = 5 5 2 4, v =, w =. 1 1 0 0 1 1 5 5 Find the following. (a) AB BA (b) Av B (2w v) (c) A (d) (w w) w (e) (Cv) (2Aw) (Av) w (f) (Cv) (Cw) v w (g) The projection of v onto w, i.e. P w (v). (h) The projection of w onto v, i.e. P v (w). (i) The projection of Av onto v, i.e. P v (Av). (150) Find the angle in degrees between the vectors p and q: (a) p = (2, ), q = ( 2, 1) (b) p = 1 2, q = 2 1 2 2 (c) p = ((1, 1, 1, 1), ) q = (2,( 1, 1, 2) ) ( ) ( ) ( ) A11 A (151) If A = 12 v1 w1 1 0, v =, w =, e A 21 A 22 v 2 w 1 =, e 2 0 2 =, prove 1 that (a) v = v 1 e( 1 + v 2 e) 2. A11 (b) Ae 1 =. A ( 21 ) A12 (c) Ae 2 =. A ( 22 ) ( ) A11 A12 (d) Av = v 1 + v A 2. 21 A 22 ( A11 A 21 (e) (Av) w = v (Bw), where B = A 12 A 22 ). (152) Find two different nonzero vectors that are perpendicular to (2,, 1). No fair picking vectors that are scalar multiples of each other. (15) Let F (x, y) = (x y, y + x 2 ). Show that F does not preserve distances between points, so that it is not an isometry.

(154) Let F (x, y) = (y 1, x + 1). Rewrite F in the matrix form ( ) ( ) ( ) x x b1 F = A +. y y b 2 Is F an isometry? Describe how the transformation works. (155) Find an isometry G : R 2 R 2 that rotates points by 60 degrees counterclockwise and then translates 4 units up. (156) For each item listed below, find the function G : R 2 R 2 of the form G (v) = Av + b that does the job. (a) All points are rotated around the origin by 45 degrees counterclockwise. (b) All points are first rotated around the origin by 45 degrees counterclockwise and then translated by the vector (, 4). (c) All points are reflected across the line through the origin that is at the angle 0 degrees (measured from the positive x axis, as usual). (157) For each item listed below, find the formula for the isometry F : R 2 R 2 that is described in words. (a) First, rotate by 0 degrees clockwise around the origin, then translate by the vector 2ı + j. (b) First, translate by the vector 2ı + j, then rotate around the origin ( 0) degrees clockwise. (Answer: with points written in vector form, F = x y ) ( x + 1y + 1 2 2 2 1x + y + + 1.) 2 2 2 (c) First, rotate by 0 degrees clockwise around the origin, then translate by the vector ( ( ) + 2) 1 ı + + 1 j. 2 (d) Why does it make sense that the answers to (a) and (b) are different, and why does it make sense that the answers to (b) and (c) are the same? (158) Find the formula for the map φ : R 2 R 2 that reflects points across the line y = x. (159) Answer the following. (a) Find the formula for an isometry which is given as the following sequence of operations: (1) Rotate by 2 degrees (counterclockwise) around the origin. (2) Rotate by 88 degrees (counterclockwise) around the origin. (b) Find the formula for an isometry which is given as rotation by 240 degrees clockwise around the origin. Why is your answer the same as in (a)? (c) Find the formula for an isometry which is given as the following sequence of operations: (1) Rotate by 0 degrees counterclockwise around the origin. (2) Reflect about the x-axis () Rotate by 0 degrees clockwise around the origin. (d) Find the formula for an isometry which is given as reflecting across the line y = 1 x. Why is your answer the same as in (c)? (160) You are given the parallelogram ABCD where A = (1, 1), B = (2, ), C = (0, 7), D = ( 1, 5). Find a specific isometry ψ : R 2 R 2 that translates and rotates the parallelogram so that ψ (A) = (0, 0) and ψ (B) is on the positive x-axis. 11

12 (161) For each item listed below, find the function G : R 2 R 2 of the form G (v) = Av + b that does the job. (a) All points are rotated around the point (1, 2) by 45 degrees clockwise. (b) All points are reflected across the line 2x y = 9. (c) All points are reflected across the line x y =, and then the points are translated along the same line by 2 units. (upward, roughly). (162) Determine the type of isometry (translation, rotation, reflection, or glide reflection). (a) g (x, y) = ( ( ) x + 4y 2, 4x + 5( y + 7) 5 5 5 x 8 ) ( ) ( ) 15 (b) A = 17 17 x 2 15 y 8 + y 1 17 17 (16) Put each function in the standard form, and then determine the type of plane isometry (translation, rotation, reflection, or glide reflection). (a) g (x, ( y) ) = (y ( 1, x + 2) ) ( ) x 4 x 2 (b) A = y 1 5 4 y + 1 (c) Does the isometry in (a) fix any point(s)? If so, find it(them). (d) Does the isometry in (b) fix any point(s)? If so, find it(them). (164) The following function is a rotation around a point of R 2. Find the point and angle of rotation. (Hint: the point is fixed). ( ) ( ) ( ) ( ) x 0.6 0.8 x 1 F = + y 0.8 0.6 y 2 (165) Find the single function G : R 2 R 2 in the form G (v) = Av + b that does the following. All points are translated up 1 unit and left 2 units, and then rotated around the origin by 45. (166) Let A, B, and C be the points A = (1, 2), B = (0, 5), and C = (5, 0). Let D, E, F be the points D = (, 6), E = (2, 1), F = (1, 8). In this problem we will be considering ABC and DEF. (a) Find the measure of ABC. (b) Prove that there exists an isometry G : R 2 R 2 that maps ABC onto DEF. (c) Prove that this isometry is a reflection about a certain line, and find the equation of that line. ( ) ( ) x x (167) Find the single function G : R 2 R 2 in the form G = A + b that y y does the following. All points of R 2 are rotated around the origin by 0 clockwise, then translated down 2 units and left units. (168) By using matrix multiplication, show that if F : R 2 R 2 is reflection across the line through (0, 0) at angle A, and if G : R 2 R 2 is reflection across the line through (0, 0) at angle B, then G F : R 2 R 2 is a rotation. Also, find the angle of rotation of G F. (169) Draw a picture of each path, and then find its length (using a calculator). (a) α (t) = (1, t 2), 0 t (b) β (t) = (cos (t), sin (t)), 0 t π 4 (c) γ (t) = (cos (t), sin (t)), 0 t π 4 (d) d (t) = (t, 2t t 2 + 1), 0 t 2 (e) E (t) = ( cos (t), sin (t), 2) t, 0 t 4π

(170) Draw a sphere and three geodesics on that sphere that intersect in one point. (171) Draw a torus and three geodesics on that torus that intersect in one point. (172) Suppose that an equilateral triangle on the sphere of radius 1 has one right angle. (a) Explain why the other two angles must also be right angles. (b) Find the area of the triangle. (c) Find the length of each side of this triangle. (d) How would the answers to (b) and (c) change if the triangle is on a sphere of radius ( 2? 1 (17) Let A = ) 1, 1, and B = ( 2,, 2 5 5 5 (a) Prove that A and B are on the unit sphere S 2. (b) Find the distance between A and B as measured in R. (c) Find the distance between A and B as measured on S 2. (d) Why does it make sense that the answers to (b) and (c) are different? (174) Determine if each of the sentences below are true or false for (i) The Euclidean plane (ii) for S 2 (iii) for H 2 (a) The sum of measures of the angles in a triangle is always at least 180 degrees. (b) Every geodesic in this space can be extended so that it is infinitely long. (c) Given two fixed points, there is exactly one geodesic connecting them. (d) Given a fixed geodesic l and a point p not on the geodesic, there exists exactly one geodesic through p that does not intersect l. (175) Given a geodesic triangle on the unit sphere with angles 170 degrees, 40 degrees, and 50 degrees: (a) Find its area. (b) What are the degree measurements of the opposite triangle (i.e. the complement of the interior of the given triangle, which is also a triangle on S 2 )? (c) Find the area of the opposite triangle, in two different ways. (176) Given a geodesic triangle on S 2 with angles 60 degrees, 90 degrees, and 90 degrees: (a) Prove that this geodesic triangle is isosceles. (b) Find its area. (c) Draw a picture of this triangle on the sphere, and determine the lengths of the sides of the triangle. (177) We are given a geodesic triangle ABC on S 2 with m A = 10, m B = 70, and m C = 70. (a) Find the area of this geodesic triangle. (b) Prove that this geodesic triangle is isosceles. (c) Would your answers to (a) or (b) change if the geodesic triangle is on a sphere of radius 10? (178) Prove that there does not exist a geodesic triangle on S 2 with two vertices that are antipodal, unless the triangle is actually a lune. (179) Find a formula for the area of a spherical pentagon in terms of its interior angle measurements. ). 1

14 (180) Below is a list of statements. In the columns, indicate if the statement is true or false (T or F) in the indicated geometric setting. No justification is necessary. Statement Euclidean plane Sphere Hyperbolic Plane The sum of the measures of the angles in a triangle is always no more than 180 degrees. If A = D, B = E, and C = F, then ABC = DEF. Given two fixed points, there exists at least one geodesic connecting them. There exist an infinite number of geodesics that are parallel to each other. The ASA triangle congruence theorem is true. For any two given geodesic lines, they intersect in at least 1 point.