JUNE 1895. (a) I. (a) Define the terms "locus" and "limit of a variable" and give an example of each. (b) Prove that two triangles are similar if their homologous sides are proportional. (c) Through a given point A within a circle draw two equal chords. [Both the construction (with ruler and compass), and also the proof, are required.] 2. (a) Prove that if each of two angles of a quadrilateral is a right angle, the bisectors of the oth er angles are either perpendic ular, or parallel, to each other. (b) Prove that if the radius of a circle is divided in extreme and mean ratio, the greater part is equal to the side of a regular inscribed decagon. [The construction is not required.] JUNE 1895. (b) One question may be omitted. Logarithmic tables should be used in calculating the answers of two questions. 1. The base of a triangle is 14 inches and its altitude is 7 inches. Find the area of the trapezoid cut off by a line 6 inches from the vertex. Express the result in square meters. 2. Find the number of feet in an arc of 40° 12′ if the radius of the circle is 0.7539 meters. 3. The length of a chord is 10 feet, and its greatest distance from the subtending arc is 2 feet 71⁄2 inches. Find the radius of the circle. 4. Find the area, and also the weight in grams, of the largest square that can be cut from a circular sheet of tin 16 inches in diameter and weighing 8.2 ounces per square foot. JUNE 1896. (a) 1. The sum of the three angles of a triangle is equal to two right angles. 2. Construct a circle having its center in a given line and passing through two given points. 3. The bisector of the angle of a triangle divides the opposite side into segments which are proportional to the two other sides. 4. If two angles of a quadrilateral are bisected by one of its diagonals, the quadrilateral is divided into two equal triangles and the two diagonals of the quadrilateral are perpendicular to each other. 5. The circumferences of two circles are to each other as their radii. (Use the method of limits.) JUNE 1896. 1. A tree casts a shadow 90 feet long, when a vertical rod 6 feet high casts a shadow 4 feet long, How high is the tree? 2. The distance from the center of a circle to a chord 10 inches long is 12 inches. Find the distance from the center to a chord 24 inches long. (Use log 3. The diameter of a circular grass plot is 28 feet. Find the diameter of a grass plot just twice as large. arithms.) 4. Find the area of a triangle whose sides are a = 12.342 meters, b = 31.456 meters, c = 24.756 meters, using the JUNE 1897. (a) 1. If two triangles have three sides of one equal to three sides of the other, each to each, the triangles are equal. 2. The straight lines joining the middle points of the adjacent sides of any quadrilateral form a parallelogram whose perimeter is equal to the sum of the diagonals of the quadrilateral. 3. To construct a circle of given radius tangent to two given intersecting straight lines. 4. If through a fixed point within a circle two chords are drawn, the product of the two segments of one is equal to the product of the two segments of the other. 5. The area of a circle is equal to half the product of its radius and circumference. |