any) you use to prove this. No marks will be given for a mere proof if this be not done. May 1876. 1. If two triangles have two sides of the one equal to two sides of the other, each to each, and have also the angles contained by those sides equal to one another, they shall also have their bases or third sides equal. 2. Any two angles of a triangle are together less than two right angles. 3. The opposite sides and angles of a parallelogram are equal to one another. 4. If two isosceles triangles have the same base the line which joins their summits is at right angles to the base. 5. ABC is an equilateral triangle. On AB take any distance AF less than AB. On BC take BD equal to AF, and on CA take CE equal to AF. Join AD, BE, and CF. Show that these three lines either meet in one point or enclose an equilateral triangle. 6. What use does Euclid make in his first book of the tenth axiom, namely that two straight lines cannot enclose a space. May 1877. I. Prove that the three angles of a triangle taken together are equal to two right angles. 2. Through a given point outside a straight line draw another straight line which shall make with the first an angle equal to a given angle. How many such lines can be drawn? 3. Prove that if the squares upon two of the sides of a triangle, taken together, are equal to a square on the third side, the triangle is right angled. Which side is the right angle opposite to? 4. Prove that if the four sides of a parallelogram are all equal, the two diagonals bisect, and are perpendicular to, one another. 5. The equal sides of an isosceles triangle are each 13 inches long, and the base is 10 inches. Find the length of the shortest line which can be drawn from the vertex to the base. 6. What do you understand by the equality of two angles? By what geometrical process would you ascertain which was the greater of two angles actually drawn upon a sheet of paper (their vertices not coinciding)? Stage II. May 1875. 1. Describe a square which shall be equal to a given rectilineal figure. 2. Let ABCDE be the corners of a regular pentagon, taken in order. Join AC and BD meeting in H. Show that AEDH is a parallelogram. 3. State and prove the relation which exists between the angle which a chord of a circle makes with a tangent and the angle in the segment. 4. Two parallel tangents to a fixed circle are met by a third tangent in P and Q. Find the angle which PQ subtends from the centre of the circle. 5. Two given circles ABP, ACQ intersect in a fixed point A. B and C are two fixed points, one in each circumference. PAQ is a moveable chord through A, and the right lines joining PB and QC meet in R. Show that the locus of R is a circle passing through B and C. 6. Two circles intersect in the points A and B. A right line intersects the chord AB in P, one circle in C and D, and the other circle in G and H. Show that the rectangle between PC and PD is equal to that between PG and PH. May 1876. 1. If a straight line be divided into two equal and also into two unequal parts, the squares on the two unequal parts are together double the square on half the line and of the square on the line between the points of section. 2. In equal circles, equal angles stand on equal arcs, whether they are at the centres or circumferences. 3. If all the sides of any polygon touch a circle, which is wholly contained within the polygon, then the area of the polygon is half the rectangle between a line equal to its perimeter and the radius of the circle. 4. Let ABCD be four points on a straight line taken in order. Show that the rectangle AC·BD is equal to the sum of the rectangles AB CD and AD BC. 5. Two circles touch one another at a point A. Any two points B and C are taken upon one of them, and the straight lines AB, and AC, produced either way if necessary, meet the other circle in D and E. Show that the straight lines B C and DE are parallel. 6. Two straight lines AC,BD passing through a point P, meet a pair of parallel lines, in the points A and B on one of the parallel lines, and in the points C and D on the other. Along PA, take PF equal to PB, and along PB take PE equal to PA. Show that a circle can be drawn through the four points CDEF. May 1877. 1. Prove that, if a straight line be divided into any two parts, the rectangles contained by the whole and each of the parts are together equal to the square on the whole 2. Prove that if two circles touch one another, their line. centres and the point of contact lie in one straight line. What different cases are there of this proposition? 3. Divide a given straight line into two parts, such that the sum of their squares shall be equal to a given square. What limitation is implied in this question as to the size of the square? 4. Prove that the diameter is greater than any other chord which can be drawn in a circle. 5 Three circles are described on the sides of a triangle as diameters. Prove that they intersect two by two on the sides of the triangle. 6. Prove that if a quadrilateral be circumscribed to a circle, the right lines which bisect its angles meet in a point. THE END. Aniversity of London. JUST PUBLISHED, Crown 8vo, Cloth extra, Price 4/ MATHEWS' MATRICULATION MATHEMATICS Being all the Papers in ARITHMETIC AND ALGEBRA Set at the MATRICULATION EXAMINATIONS of the LONDON UNIVERSITY, from 1844 to 1878 (inclusive), with Complete Answers to all the Questions, Solutions in Arithmetical & Geometrical Progression, AND Hints for Private Students. BY THE AUTHOR OF Deductions from Euclid and How to Work Them, etc. N.B.-This Book contains nearly 1,500 Examples in the leading rules of Arithmetic and Algebra, all set by the highest Mathematical Authorities, and will be valuable not only to Matriculation Students, but to Students in Training Colleges, the Upper Forms of Public Schools, Pupil Teachers, and others. Copies post free to Teachers and Students for 3s. 6d., P.O.O. (preferred) or stamps. E. H. MATHEWS, 192, Queen's Rd., Peckham, S.E. |