APPENDIX B. GEOMETRY PAPERS, SET AT THE Matriculation Examinations of the London 1874-77. MATRICULATION, JANUARY 1874. Afternoon, 3 to 6. GEOMETRY. [Each Candidate is desired to state the Text-Book of Geometry used by him in preparing for this Examination.] [The only abbreviations which can be permitted are sq. for "square," rect. for "rectangle," ||gram. for "parallelogram," L for "angle," and the symbols.,,=, and in their usual senses.] 1. Define a Circle, a Parallelogram, an Oblong, a Rhombus, a Rhomboid, and Parallel straight lines. State also the Postulates of Euclid, or the processes regarded as executable and allowable in ordinary geometry. Would it be allowable in ordinary geometry to make the two ends of a given straight line simultaneously describe circles about two different centres ? 2. "If a straight line falling on two other straight lines make the alternate angles equal to each other, these two straight lines shall be parallel." What supposition is tacitly assumed concerning the "two straight lines" in this (the ordinary) enunciation? Why is the proposition not necessarily true as it stands ? Having completed the enunciation, prove the proposition. 3. Prove that the three angles of a triangle are together equal to two right angles; and investigate the sum of the exterior angles of any plane convex polygon. 4. Show how to bisect any given rectilineal angle. Show also how to trisect a right angle. 5. Prove that in any right-angled triangle the square upon the hypothenuse is the sum of the squares upon the two sides. 6. Divide a given straight line into two parts so that the rectangle contained by the whole and one of the parts shall be equal to the square on the other part. 7. State the relation which connects the side opposite the obtuse angle of an obtuse-angled triangle, the sides containing the obtuse angle, and the segment lying in either one of these sides produced and intercepted between the perpendicular drawn from the opposite acute angle upon the side so produced and the obtuse angle. Let ABC be any obtuse-angled triangle, C the obtuse angle, AD, BE perpendiculars upon BC, AC produced and meeting these produced lines in D and E: prove that the rectangle of BC, CD is equal to that of AC, CE. 8. Prove that the angles in the same segment of a circle are equal to one another. 9. Draw a tangent to a circle from any point without it. 10. If two straight lines cut one another within a circle, prove that the rectangle contained by the segments of the one is equal to the rectangle contained by the segments of the other. Extend this theorem, but without proof, to the case of two lines meeting in a point external to the circle, and each of them either cutting or touching the circle. II. Inscribe a circle in a given triangle. Find a point equidistant from each of three straight lines in a plane which do not coincide in direction with the sides of any triangle that can be drawn in the plane. Is the construction required in this last problem always possible? MATRICULATION, JUNE 1874. Afternoon, 3 to 6. GEOMETRY. 1. Bisect the angle between two unlimited straight lines which intersect. Through a given point draw a straight line equally inclined to two given straight lines. How many such lines may be drawn? 2. Prove that the diagonals of the parallelogram bisect each other. Prove also, conversely, that a quadrilateral is a parallelogram if the diagonals bisect each other. 3. Prove that the angle at the vertex of a triangle is less than, equal to, or greater than a right angle, according as the line joining the vertex to the middle point of the base is greater than, equal to, or less than half the base. 4. On a given base construct a rectangle equal in area to a given triangle. 5. The hypothenuses of three isosceles right-angled triangles form a right-angled triangle. Construct the figure; and prove that one of the given triangles is equal to the sum of the other two. 6. If a straight line be divided into any two parts, the squares of the whole line and of one of the parts are equal to twice the rectangle contained by the whole and that part, together with the square of the other part. 7. Prove that a straight line cannot have more than two points in common with the circumference of a circle. What relation must exist between the radius of the circle and the distance of the centre from the straight line in order that the line may have two points, or one, or none, in common with the circle? 8. Through three given points not in the same straight line one circle may always be described, and only one. 9. In the same circle all angles at the circumference standing upon the same arc, or on equal arcs, are equal. 10. Having given an equiangular and equilateral pentagon inscribe a circle in it. MATRICULATION, JANUARY 1875. Afternoon, 3 to 6. GEOMETRY. 1. Construct an isosceles triangle whose base and angle at the vertex are given. 2. The greater side in every triangle is opposite to the greater angle; and, conversely. The greater angle of every triangle is opposite to the greater side. 3. If through the middle point of a side of a triangle a straight line be drawn parallel to the base, it will bisect the other side, and that part of it which is intercepted between the sides of the triangle will be equal to half the base. 4. If A be the point of intersection of the diagonals of a parallelogram, then every straight line drawn through A will meet opposite sides of the parallelogram in points equidistant from A, and will divide the parallelogram into two equal parts. |