cushion of the table at the same angle at which it strikes, find the point at which a player should aim his ball if he wishes to strike another ball after one rebound. Prove the correctness of your construction. 2. Prove that, if two triangles have an angle of one equal to an angle of the other and the including sides proportional, the triangles are similar. Show that the middle points of the diagonals, and of the non-parallel sides of a trapezoid lie in line. 3. Compute to three significant figures the length of the shortest piece of wire which will just go around two pipes lying side by side, the diameters being 2 inches and 6 inches respectively. 4. Four rods AB, BP, CQ, QD are pivoted together so that BC=DQ, BD= CQ. The point A is fixed. Show that if AQP are once in line, they will remain in line when P moves freely about the plane, and A show that AQ will bear а constant ratio to AP. (This instrument is called a pantograph.) P 5. A point P lies within a fixed circle. Find the locus of points which lie one-third of the way from P to the points of the circumference. JUNE, 1909 (Two hours.) The University provides a Syllabus. 1. Show that if a triangle be equilateral, it is also equiangular. Is this theorem true in the case of a quadrilateral? Give your reason. 2. Prove that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides. Deduce from this a proof that of two oblique lines drawn from the same point of a perpendicular and cutting off unequal distances from the foot of the perpendicular, the more remote is the greater. 3. Prove that if the products of the segments into which the diagonals of a quadrilateral divide one another are equal, a circle may be circumscribed about the quadrilateral. 4. A square 10 inches on a side is changed into a regular octagon by cutting off the corners. Find the area of this octagon. 5. A wheel 40 inches in diameter has a flat place 5 inches long on the rim. Describe carefully the locus of the centre as the wheel rolls along the level. SEPTEMBER, 1909 (Two hours.) The University provides a Syllabus. Takе л= 22. 1. Prove that the opposite sides and the opposite angles of a parallelogram are equal. Prove that the only sort of a parallelogram that can be inscribed in a circle is a rectangle. 2. Show, with proof, how to inscribe a circle in a given triangle. Explain the difference between this problem and that of constructing a circle to touch three given lines. 3. Show that the areas of two similar triangles are to each other as the squares of any two homologous sides. Show that the quadrilateral formed by joining the middle points of the adjacent sides of a given quadrilateral is a parallelogram whose area is one-half that of the quadrilateral. 4. At two points A and B of the circumference of a circle whose centre is O, tangents are drawn. The points A' and B' lie respectively on these tangents, and the angle OA'B' is a right angle. Prove that BB'2=AA'2+A'B'2. 5. A circular basin of radius r is full of water, and upon the surface there floats a thin straight stick of length 14. Find the area of that region of the surface which is inaccessible to the middle point of the stick. JUNE, 1910 (Two hours.) The University provides a Syllabus. 1. Prove that if the sides of one angle be perpendicular respectively to those of another, the angles are either equal or supplementary. 2. Show how to inscribe a circle in a given triangle. How many circles can be drawn to touch three given lines? Are there any positions of these lines for which the number is less? 3. Define" incommensurable magnitudes." Give a careful proof of some theorem where such magnitudes occur. 4. ABCD are the vertices in order of a quadrilateral which is circumscribed to a circle whose centre is O. Prove that AOB and COD are supplementary. 5. Two radii of a circle OA and OB make a right angle. A second circle is described upon AB as diameter. Prove that the area of the crescent-shaped region outside of the first circle, but inside of the second, is equal to that of the triangle AOB. (Hippocrates, fifth century B. C.) SEPTEMBER, 1910 The University provides a Syllabus. 1. Prove that if two equal oblique lines be drawn from a point to a line they must meet it at equal distances from the foot of the perpendicular. Prove that a straight line cannot intersect a circumference more than twice. 2. Prove that if two circles intersect, the common chord is perpendicular to the line of centres. 3. Prove that in a right triangle the perpendicular on the hypotenuse is a mean proportional between the segments of the hypotenuse. In a right triangle ABC, whose right angle is at B, P is the foot of the perpendicular from B upon the hypotenuse AC, and BP produced intersects in D the perpendicular to AB at A. Prove that PA×PB=PC×PD. (Plato, fourth century B. C.) 4. A square is rolled along a straight line without slipping. What is the length of the locus of one corner as the square makes one complete turn? 5. A regular hexagon is reduced to one of smaller size by cutting straight across from the middle point of each side to that of the next. What proportion of the area is removed? EXAMINATIONS IN PLANE GEOMETRY FOR ADMISSION TO YALE JUNE, 1889 (Time allowed, one hour.) 1. Solve one of the two following problems with ruler and compass. No explanation is to be written, but each step of the construction is to be made clear on the figure. (a) Construct a right-angled isosceles triangle and inscribe a circle in it. (b) Divide a straight line AB into three equal parts. Erect a square on the middle part and construct a triangle of equal area. 2. From two points A and B, on an arc of a circle straight lines, AC, AD, BC, BD, are drawn to the ends of the chord of the arc. Find two similar triangles in the figure, prove them similar, and write the proportions of their homologous sides. 3. Prove that in equal circles two incommensurable arcs have the same ratio as the angles which they subtend at the centre. 4. Prove that two triangles having a common angle. are to each other as the products of the sides including the common angle. 5. Solve or prove one of the following propositions: (a) To circumscribe about a circle a regular polygon similar to a given inscribed polygon. |