JUNE 1891. 1. Prove that two straight lines which are both parallel to a third straight line are parallel to each other. 2. Prove that if from any point without a circle there be drawn a line tangent to the circle and also a line cutting it and not passing through the center, the tangent is a mean proportional between the whole secant and its external segment. 3. Construct an equilateral triangle whose perimeter shall be 12 inches, and inscribe in it a circle. Note. Write nothing in connection with this problem. 4. Prove that if the points of tangency of the above inscribed circle be joined, the triangle formed is equilateral, and its perimeter is 6 inches. 5. (a) Define a geometrical locus, and give examples of loci from the figure drawn in number 3. (b) Define the limit of a variable, and write the caption of some proposition whose proof depends on the theory of limits. 121 JUNE 1892. 1. Construct accurately by ruler and compass a parallelogram ABCD having the angle A 45°, the side AB 6 units in length and the altitude 3 of the same units. Calculate the length of AC. 2. (a) State the converse of the following proposition: If a triangle is isosceles and if a straight line is drawn through the vertex parallel to the base, it bisects an exterior angle of the triangle. (b) Prove the converse as you have stated it. Make the demonstration as full and clear as possible. 3. Prove two of the following propositions. The work may be limited to drawing a figure and giving a synopsis of the demonstration. (a) If the area of a regular polygon is equal to the product of the perimeter by one-half the apothem, it follows that the area of a circle Ti R2. (b) If two lines are drawn through the same point across a circle, the products of the two distances on each line from this point to the circumference are equal to each other. (c) If the radius of a circle be divided in mean and extreme ratio, the greater segment is equal to one side of a regular inscribed decagon. 16 JUNE 1893. 1. Prove that if the diagonals of a quadrilateral bisect each other the figure is a parallelogram. 2. Prove that in any right-angled triangle the square on the side opposite to the right angle is equal to the sum of the squares on the other two sides. A purely geometrical proof is preferred. State fully each principle employed in the proof. 3. Given a straight line AB, of indefinite length, and a point C without it. Find a point in AB equally distant from A and C. Make the necessary construction accurately with ruler and compass. In what case is the solution impossible? 4. Given an angle COD at the center of a circle and the line CA meeting DO produced in A so that AB is equal to the radius of a circle. Prove that the angle A is equal to one-third of the angle COD. JUNE 1894. (a) 1. If the diagonals of a quadrilateral bisect each other, the figure is a parallelogram. 2. To draw a tangent to a given circle, so that it shall be parallel to a given straight line. 3. If AB is a chord of a circle, and CE is any chord drawn through the middle point C of the arc AB cutting the chord AB at D, prove that the chord AC is a mean proportional between CD and CE. 4. The areas of two similar triangles are to each other as the squares of any two homologous sides. 5. The area of a circle is equal to one-half the product of its circumference and radius. JUNE 1894. (b) 1. What is the number of degrees in each angle of a regular decagon? 2. Find the area in square feet of an equilateral triangle whose side is 3 meters. 3. ABC is a right triangle. The sides AC and BC about the right angle C are respectively 50 and 120 feet. Divide the triangle into 2 parts equal in area by a line DF parallel to BC. Compute the length of the three sides of the triangle ADF. 4. The area of a circle is a hectare. What is its diameter. 5. Calculate in meters the length of a degree on the circumference of the earth, assuming the section of the earth to be a circle whose radius is 3963 miles. [Those taking the preliminary examinations must use logarithms.] [For preliminary candidates only.] 6. Find the value of the following expression by loga3/(.06342)3 X 187.32 rithms: .34216 X 6.0372 |