Faculty of Engineering, Mathematics and Science School of Mathematics

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1 Faculty of ngineering, Mathematics and Science School of Mathematics Sample Paper Trinity Term 2016 M232: uclidean and non-uclidean Geometry Prof. avid Wilkins Instructions to andidates: redit will be given for the best 3 questions answered. Materials Permitted for this xamination: Formulae and Tables are available from the invigilators, if required. Non-programmable calculators are permitted for this examination, please indicate the make and model of your calculator on each answer book used. You may not start this examination until you are instructed to do so by the Invigilator. Page 1 of 12

2 1. The SS ongruence Rule and the Isosceles Triangle Theorem (Pons sinorum) included in the Selected Propositions from uclid at the end of this examination paper may be used without proof throughout this question. (a) (uclid, ook I, Proposition 16, John asey s dition) F G Prove that if any side () of a triangle () be produced, the exterior angle () is greater than either of the two interior non-adjacent angles. (Thus the task is to prove that the external angle is greater than the interior angles of the triangle at and.) The proof must not rely on later propositions in uclid. In particular the proof must not make use of the results of ook I, Proposition 32 that assert that the interior angles of the triangle sum to two right angles and the exterior angle is the sum of the two interior non-adjacent angles at and. Note that the construction underlying the above diagram ensures that the point bisects both and F. (6 points) (Question continues on the next page) Page 2 of 12 c TRINITY OLLG ULIN, TH UNIVRSITY OF ULIN 2016

3 (b) (uclid, ook I, Proposition 18, John asey s dition) Prove that, if in any triangle () one side () be greater than the other (), the angle oppositie to the greater side is greater than the angle opposite to the less. Note that, in the above figure, is greater than and is equal to. It must be shown that the angle of at is greater than that at. (4 points) (c) Prove that, if one angle () of a triangle () be greater than another angle (),a the side () which is opposite to the greater angle is greater than the side () which is opposite to the less. (2 points) (Question continues on the next page) Page 3 of 12 c TRINITY OLLG ULIN, TH UNIVRSITY OF ULIN 2016

4 (d) (uclid, ook I, Proposition 20, John asey s dition) Prove that, the sum of any two sides (, ) of a triangle () is greater than the third. Note that, in the above diagram, the points, and are colinear, and the line segments and are equal. (4 points) (e) (uclid, ook I, Proposition 21, John asey s dition) Prove that, if two lines (, ) be drawn to a point () within a triangle from the extremities of its base (), their sum is less than the sum of the remaining sides (, ), but they contain a greater angle. (4 points) (nd of Question) Page 4 of 12 c TRINITY OLLG ULIN, TH UNIVRSITY OF ULIN 2016

5 2. The Selected Propositions from uclid at the end of this examination paper may be used without proof throughout this question. In addition you may use, without proof, the result of uclid, ook I, Proposition 24, which ensures that if in two triangles and F the side and are equal to sides and F respectively, and if angle is greater than angle F then is greater than F. You may also use, without proof, any of the results stated in Question 1. (a) (uclid, ook III, Proposition 3, John asey s dition) O (i) Prove that if a line () passing through the centre of a circle bisects a chord (), which does not pass through the centre, then it cuts it at right angles. (ii) Prove that if a line () passing through the centre of a circle cuts at right angles a chord (), which does not pass through the centre, then it bisects it. (8 points) (Question continues on the next page) Page 5 of 12 c TRINITY OLLG ULIN, TH UNIVRSITY OF ULIN 2016

6 (b) (uclid, ook III, Proposition 7, adapted from John asey s dition) O P G F From any point P within a circle, which is not the centre O, let be the diameter of the circle that passes through the points O and P, so that the points, O, P and occur in that order along the diameter and the points and lie on the circumference of the circle. Let, and be points around the circumference of the circle between and located so that the points,,, and are distinct and occur in that order around the circumference of the circle, as depicted in the above figure. (i) Prove that P is greater than P, and that P is greater than P. (ii) xplain why there exists exactly on point F on the circumference of the circle distinct from the point for which the line segments P and P F are equal. (6 points) (Question continues on the next page) Page 6 of 12 c TRINITY OLLG ULIN, TH UNIVRSITY OF ULIN 2016

7 (c) (uclid, ook III, Proposition 11, John asey s dition) P H O Prove that if one circle (P ) touch another circle (P ) internally at any point P, the line joining the centres must pass through that point. (6 points) (nd of Question) Page 7 of 12 c TRINITY OLLG ULIN, TH UNIVRSITY OF ULIN 2016

8 3. (a) Let u, v, w and z be vectors in R 3. Prove the following identity (known as Lagrange s Quadruple Product Identity): (u v). (w z) = (u. w)(v. z) (u. z)(v. w). (b) Let r, r and r be vectors in R 3. Prove that (5 points) (r r ) (r r ) = (r.(r r ))r. (5 points) (c) Let L, L and L be three distinct points on the unit sphere that do not all lie on any one great circle of the sphere, and let p be the angle which the line from the centre of the sphere to the point L makes with the plane through the centre of the sphere that contains the points L and L. Prove that sin p = sin L sin LL = sin L sin L L, where sin L denotes the sine of the angle between the arcs LL and LL at L and sin L denotes the sine of the angle between the arcs L L and L L at L. (10 points) [Note: throughout the above question, you may use, without proof, the Vector Triple Product Identity, which asserts that u (v w) = (u. w) v (u. v) w for all vectors u, v, w. You may also use, without proof, standard properties of the Scalar Triple Product.] (nd of Question) Page 8 of 12 c TRINITY OLLG ULIN, TH UNIVRSITY OF ULIN 2016

9 4. Let Σ be a smooth surface in R 3, and let p and q be smooth local coordinates defined throughout Σ. Let the partial derivatives of the artesian coordinate functions with respect to these smooth local coordinates p and q be denoted by a, b, c, a, b and c, so that x p = a, y p = b, z p = c, x q = a, y q = b, z q = c. (a) Show that the unit normal vector field (X, Y, Z) along the surface Σ satisfies where (X, Y, Z) = ±1 (bc cb, ca ac, ab ba ), = (bc cb ) 2 + (ca ac ) 2 + (ab ba ) 2 ). (b) Using the notation of (a), prove that (p, q) = ±1 Z(p, q) ( x y p q x q ) y. p (5 points) (5 points) (Question continues on next page) Page 9 of 12 c TRINITY OLLG ULIN, TH UNIVRSITY OF ULIN 2016

10 (c) Let ν: Σ S 2 denote the Gauss map of the smooth surface Σ, defined such that ν(p, q) = (X(p, q), Y (p, q), Z(p, q)) where X(p, q), Y (p, q) and Z(p, q) are the artesian coordinates of the unit normal vector field to the surface at a point with local coordinates p and q. Prove that area(ν(r)) = R k dσ, where R is a region with regular boundary contained within the domain of a smooth local coordinate system (p, q) on the surface, the integral is taken with respect to surface area, and k(p, q) = X Y p q X Y q p x y p q x. y q p (You may use without proof the result that dσ = (p, q) dp dq, where is defined as specified in (a).) (10 points) (nd of Question) Page 10 of 12 c TRINITY OLLG ULIN, TH UNIVRSITY OF ULIN 2016

11 Selected Propositions from uclid SS ongruence Rule (see uclid, ook I, Proposition 4) F If two triangles (, F ) have two sides (, ) of one equal respectively to two sides (, F ) of the other, and also have the angles (, ) included by those sides equal, the triangles shall be equal in every respect that is, their bases or third sides (, F ) shall be equal, and the angles (, ) at the base of one shall be respectively equal to the angles (, F ) at the base of the other; namely, those shall be equal to which the equal sides are opposite. Isosceles Triangle Theorem (Pons sinorum) (see uclid, ook I, Proposition 5) The angles (, ) at the base () of an isosceles triangle are equal to one another, and if the equal sides (, ) be produced, the external angles (, ) below the base shall be equal. onsider the following configuration: Page 11 of 12 c TRINITY OLLG ULIN, TH UNIVRSITY OF ULIN 2016

12 G H F Suppose that the line F cuts the lines and as shown in the figure, intersecting at G and at H. The angles GH and G are said to be vertically opposite angles, the angles GH and GH are said to be alternate angles, and the angles G and GH are said to be corresponding angles. Vertically Opposite ngles Theorem (see uclid, ook I, Proposition 15) Let and H be lines in the uclidean plane that intersect at the point G. Then GH = G. (Thus vertical angles are equal.) lternate ngles Theorem (see uclid, ook I, Propositions 27 and 29) Let, and F be lines in the uclidean plane, where and lie on the same side of F, and let the line F cut the lines and at G and H respectively (as depicted above). Then the lines and are parallel if and only if GH = GH. (Thus and are parallel if and only if alternate angles GH and GH are equal.) orresponding ngles Theorem (see uclid, ook I, Proposition 29) Let, and F be lines in the uclidean plane, where and lie on the same side of F, and let the line F cut the lines and at G and H respectively (as depicted above). Then the lines and are parallel if and only if G = GH. (Thus and are parallel if and only if corresponding angles G and GH are equal.) Page 12 of 12 c TRINITY OLLG ULIN, TH UNIVRSITY OF ULIN 2016