SEPTEMBER 1894. PLANE. 1. Define equal polygons, equivalent polygons, regular polygons, similar polygons, a maximum polygon, a minimum polygon. 2. If a chord of a circle is greater than another chord, it is nearer the center. State the converse proposition and prove it. 3. The sum of the squares on the four sides of any quadrilateral equals the sum of the squares on the diagonals and four times the square on the line joining middle points of diagonals. 4. Construct a triangle and a square each of which is equivalent to a given pentagon. Give proof of constructions. 5. If the perimeter and base of a triangle are given, when is the area greatest? JUNE 1894. SOLID. 1. Define a polyhedron, equal polyhedrons, equivalent polyhedrons, regular polyhedrons, similar polyhedrons, symmetrical spherical triangles, polar spherical triangles. 2. What is the locus of all points equidistant from the faces of a dihedral angle? Prove your answer. 3. (a) Volumes of rectangular parallelopipeds are as the products of their three dimensions. (b) The volume of any rectangular parallelopiped is the product of its three dimensions. 4. Show that, in a regular tetrahedron, the opposite edges are perpendicular to each other. 5. Two spheres have radii of 5 and 12 inches respectively, and their centers are 13 inches apart; find the area of that portion of the surface of either sphere which is outside the other. SEPTEMBER 1894. SOLID. 1. Define a plane surface, a cylindrical surface, a conical surface, a spherical surface, a prism, a regular prism, a pyramid, a regular pyramid, a right cylinder, a right cone. 2. (a) If two planes are perpendicular, a line in one of them perpendicular to their intersection is perpendicular to the other. (b) If two planes are perpendicular, a line drawn through any point of their line of intersection perpendicular to one plane will lie in the other plane. 3. Give the theorem and proof on the area of a frustum of a regular pyramid. Give the theorem on the volume of a frustum of any pyramid. 4. A sphere circumscribes a cube whose edge is 5 inches; get the surface and volume of the sphere, and compare them with the surface and volume, respectively, of the cube. JUNE 1895. PLANE. 1. If two sides of a triangle are unequal, the angles opposite are unequal, and the greater angle is opposite the greater side. Also state and prove the converse proposition. 2. Define a locus. Find the locus of the centers of all circles whose circumferences pass through two given points. 3. An angle formed by two tangents intersecting without the circumference is measured by one-half the difference of the intercepted arcs. Give values of the arcs when the tangents are perpendicular to each other. 4. Construct a circle which shall be tangent to a given line and touch a given circle in a given point. 5. A circle may be circumscribed about any regular polygon; and a circle may also be inscribed in it. SEPTEMBER 1895. PLANE. 1. Define an angle, equal angles, oblique angles, complementary angles, supplementary angles, an inscribed angle, an angle inscribed in a segment, a re-entrant angle, the unit of angle. 2. The sum of two perpendiculars dropped from any point in the base of an isosceles triangle to the equal sides is constant and equal to the perpendicular let fall from the vertex of one of the equal angles to the opposite side. 3. The area of a trapezoid is equal to the product of its altitude by half the sum of its parallel sides. 4. Construct a square equivalent to a given parallelogram. 5. Two regular polygons of the same number of sides are similar. If S and S' are their areas, what is the ratio of the circumferences of their circumscribed circles? |