Higher Geometry Problems

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1 Higher Geometry Problems ( 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 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. (5 If we consider a finite set S of points in the plane, explain why the number of points of S is a geometric quantity. (6 Give the precise definition of a triangle. (7 Give the precise definition of a quadrilateral. (8 Give the precise definition of a rectangle. (9 Give the precise definition of a parallelogram. (0 Give the precise definition of a circle in the plane. (Note: a circle does not include its insides. ( Give the precise definition of a disk in the plane. (2 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. ( Write down a statement and proof of the formula for the area of a parallelogram. (4 Write down the precise definition of equality of angles. (5 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. (6 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. (7 Prove the formula for the area of a triangle. (8 Give the precise definition of an isosceles triangle and the precise definition of an equilateral triangle. (9 Prove that if triangle ABC is isosceles with AB = AC, then B = C. (20 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. (2 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. (22 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. (2 Explain how to construct the center of a given circle. Prove that your construction works. (24 Explain how to construct the diameter of a given circle that contains a given point on the circle. Prove that your construction works.

2 2 (25 Prove the SAS triangle congruence theorem, using basic notions only. (26 Prove the ASA triangle congruence theorem. You may assume that the SAS triangle congruence theorem has been proved. (27 Suppose that l 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 and l 2 must intersect l. [Make sure that your argument is short and concise.] (28 Prove that if L, L 2, and L are three lines in the plane, then if L L 2 and L 2 L then L L. (29 Prove the SAA triangle congruence theorem. You may assume that the ASA triangle congruence theorem has been proved. (0 Prove the SSS triangle congruence theorem. You may assume that the SAS triangle congruence theorem has been proved. ( Prove that there is no SSA triangle congruence theorem. (Hint: give a counterexample. (2 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. ( 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. (4 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. (5 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. (6 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. (7 Show how to bisect an angle using a straight edge and compass construction. Prove that your construction works. (8 Prove that a simple quadrilateral is a parallelogram if and only if its opposite sides are congruent. (9 Demonstrate that the word simple cannot be deleted from the previous exercise. (40 Prove that in a parallelogram, opposite angles are congruent. (4 Prove or disprove that the diagonals of a parallelogram bisect each other. (42 Prove or disprove that the diagonals of a parallelogram are congruent.

3 (4 Prove or disprove that if the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. (44 Prove or disprove that if the diagonals of a quadrilateral are congruent, then the quadrilateral is an isosceles trapezoid. (45 Prove or disprove that if the diagonals of a trapezoid are congruent, then the quadrilateral is an isosceles trapezoid. (46 Define kite. (47 Prove or disprove that a quadrilateral is a kite if the diagonals are perpendicular. (48 Prove or disprove that if a quadrilateral is a kite, then the diagonals are perpendicular. (49 Prove that the number n is constructible, if n is any positive integer. (50 Prove that 75 is irrational. (5 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. (52 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 = QR. Construct the line segments P S and P T. Prove that P ST is an isosceles triangle. (5 Let ABC be a triangle such that the perpendicular bisector of AB intersects BC at its midpoint. Prove that ABC is a right triangle. (54 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 80 and less than 60. (55 Prove that 5 is irrational. (56 Give two different proofs of the Pythagorean Theorem. (57 Given any quadrilateral ABCD, prove that the midpoints of the sides are either collinear or form the vertices of a parallelogram. (58 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. (59 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. (60 Find the converse of each statement above, and determine if it is true or false. Provide proof. (6 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.

4 4 (62 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. (6 An isosceles right triangle has side lengths x and x + 2. Find x. Justify the steps. (64 Suppose that a triangle ABC has points P on AB, Q on BC, and R on AC. Suppose that AP = BQ = x > and BP = QC = CR =. 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. (65 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. (66 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 =, Y Q =.5, and P, Q, and R are collinear. Find ZR. (67 State the converse to Ceva s Theorem. (68 Prove that, given three points of the plane that are not colinear, there exists a unique circle that contains the three points. (69 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. (70 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 the height of A above BC. (7 Using analytic geometry, prove that the perpendicular bisectors of any triangle are concurrent. (72 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. (7 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. (74 Give the degree measure of each angle. (a π 5 (b.24 (75 Give the radian measure of each angle. (a 80 (b 2.45

5 (76 Find the angle in radians and degrees corresponding to the location on the unit circle. (Note: give all possible angle measurements. 5 (77 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 = π. 6 (78 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 + = (79 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 = cos (A (80 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 (8 Solve the equation sin (θ = exactly, using the unit circle (not calculator. 2 (Note that all solutions θ should be given. (82 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 + (8 Find (no calculator: (a arcsin ( (b arctan ( ( (c arcsec 2 (84 Complete the following trigonometry identities: (a cos 2 (A + = (b cos (2θ =

6 6 (c cos (2θ = (different from last answer (d cot 2 (φ + = (85 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 (86 Simplify and factor the result. Assume all variables are positive and that the expressions are defined. csc(c sec(c tan(c (a sin(c cos(c cot(c (b tan (B π cos (π + B sin (B (87 By examining graphs find the smallest positive number B so that cos (θ = sin (θ + B. (Assume θ is measured in radians. (88 A triangle has sides of length 2, 7, and 6. Using trigonometry and your calculator, find the angles: (a angle opposite 6 side =? (b angle opposite 7 side =? (c angle opposite 2 side =? (89 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 (90 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. (9 Write down a precise statement and proof of the Law of Sines. (92 Write down a precise statement and proof of the Law of Cosines. (9 Find the following, and draw a picture that demonstrates that your answer agrees with intuition. (a ı + 5j (b (, 2 + ( 5, 2 (c (, 2 ( 5, 2 (d ( ı + 2j ( 2ı + j (e (, ( , 5 5 (f ( ı + 2j ( ı 2j (94 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.] (95 Find the set of all vectors v such that v (, = 0. Draw a picture that supports your conclusion. (96 Find the set of all vectors w such that w (, = 2. Draw a picture that supports your conclusion.

7 (97 Perform ( the following ( matrix operations, if possible: (a ( ( (b ( ( ( (c ( ( 2 x (d 7 y ( ( 2 0 x (e 0 7 y ( ( ( 0 x 2 (f + 0 y 5 (g ( (h ( (i ( 2 6 ( 2 9 (98 Prove( this particular example ( of the associative ( property of matrix multiplication. If A A A = 2 B B, B = 2 x, x =, prove that (AB x = A (Bx. A 2 A 22 B 2 B 22 x ( ( ( ( ( (99 Let A =, B =, C = , v =, w = 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. (00 Find the angle in degrees between the vectors p and q: (a p = (2,, q = ( 2, (b p = 2, q = (c p = (,,,, q = (2,,, 2 7

8 8 (0 If A = ( A A 2 ( ( ( A 2 v w, v =, w =, e A 22 v 2 w = 2 0 that (a v = v e( + v 2 e 2. A (b Ae =. A ( 2 A2 (c Ae 2 =. A 22 ( ( A A2 (d Av = v + v A 2. 2 A 22 ( A A (e (Av w = v (Bw, where B = 2 A 2 A 22. ( 0, e 2 =, prove (02 Find two different nonzero vectors that are perpendicular to (2,,. No fair picking vectors that are scalar multiples of each other. (0 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. (04 Let F (x, y = (y, x +. Rewrite F in the matrix form ( ( x x F = A + y y ( b b 2. Is F an isometry? Describe how the transformation works. (05 Find an isometry G : R 2 R 2 that rotates points by 60 degrees counterclockwise and then translates 4 units up. (06 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. (07 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 + y x + y (c First, rotate by 0 degrees clockwise around the origin, then translate by the vector ( ( + 2 ı + + 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? (08 Find the formula for the map φ : R 2 R 2 that reflects points across the line y = x.

9 (09 Answer the following. (a Find the formula for an isometry which is given as the following sequence of operations: ( 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: ( 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 = x. Why is your answer the same as in (c? (0 You are given the parallelogram ABCD where A = (,, B = (2,, C = (0, 7, D = (, 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. ( 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 (, 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. (2 Determine the type of isometry (translation, rotation, reflection, or glide reflection. (a g (x, y = ( ( x + 4y 2, 4x + 5( y x 8 ( ( 5 (b A = 7 7 x 2 5 y 8 + y 7 7 ( 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 x F = + y y 2 (4 Draw a picture of each path, and then find its length (using a calculator. (a α (t = (, 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 +, 0 t 2 (e E (t = ( cos (t, sin (t, 2 t, 0 t 4π (5 Draw a sphere and three geodesics on that sphere that intersect in one point. (6 Draw a torus and three geodesics on that torus that intersect in one point. (7 Suppose that an equilateral triangle on the sphere of radius 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? (8 Let A =,, and B = ( 2,, (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.. 9

10 0 (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? (9 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 80 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. (20 Given a geodesic triangle on the unit sphere with angles 70 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.

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