SHEFFIELD SCIENTIFIC SCHOOL. GEOMETRY. JUNE 1885. PLANE. I. Prove that if in any triangle one side be greater than another, the angle opposite that side will be greater than the angle opposite the other. 2. Determine how many sides the polygon has, the sum of whose interior angles is equal to the sum of its exterior angles. Explain your method. 3. (a) How does the perpendicular from the center of a circle upon a chord divide the chord and the arc it subtends? (b) How is an angle inscribed in a circle measured? (c) How is each angle between two intersecting chords of a circle measured? (d) What is the locus of the center of a circle whose circumference passes through two given points? Give proof of your answer. 4. Divide a given finite straight line into any given number of equal parts and prove your construction. 5. Given two arcs of circles of 30° whose radii are 1 ft. and 2 ft. respectively; compare their lengths and the areas of the corresponding sectors; and calculate the length of the first and the area of its corresponding sector. 6. If two sides of a triangle be given, what angle will these sides form when its area is a maximum? Give proof. JUNE 1885. SOLID AND SPHERICAL 1. If a line intersect a plane it makes a less angle with its projection on the plane than with any other line in the plane passing through the point of intersection. 2. The sum of any two face angles of a trihedral angle is greater than the third. 3. Define a regular polyhedron. How many regular polyhedrons are there? Give their names and define each of them. 4. All lines tangent to a sphere from the same external point are equal and touch the sphere in a circle of it. 5. Define a conical surface and a cylindrical surface. 6. The volume of a cone is V ; what is the volume of a similar cone whose surface is n times as great? SEPTEMBER 1885. 1. Prove that two triangles are equal when three sides of one are equal respectively to three sides of the other. 2. A straight line cannot intersect the circumference of a circle in more than two points. 3. Upon a given straight line to describe a segment which shall contain a given angle. 4. Define similar polygons and prove that the perimeters of two similar polygons are to each other as any two homologous sides. 5. The height of a room is ten feet; how can a point in the floor directly under a given point in the ceiling be determined with a twelve foot pole? 6. If two straight lines are cut by three parallel planes, their corresponding segments are proportional. 7. Find the volume of a regular tetrahedron whose edge is E. 8. Find the area of a spherical triangle whose angles are 65°, 75°, and 80°; the radius of the sphere of which it forms a part being 10 feet. JUNE 1886. PLANE. 1. Prove that the three perpendiculars from the vertices of a triangle to the opposite sides meet in a point. 2. Two parallel chords or secants intercept equal arcs on a circumference. 3. The base of a triangle and the angle opposite being given, to construct the locus of the vertex of the latter. 4. Define similar polygons. Prove that two triangles are similar when they are mutually equiangular. 5. Determine the ratio of the area of the segment of a circle whose arc is 60° to the area of the corresponding sector. |