EXAMINATIONS IN PLANE GEOMETRY FOR ADMISSION TO WORCESTER POLY TECHNIC INSTITUTE JUNE, 1909 9 to 10.30 A.M. (NOTE. Give reasons in full for any statement made in the course of a demonstration.) 1. Define: (a) A regular polygon. (b) A rhombus. (c) A rhomboid. (d) An apothem of a polygon. (e) A secant of a circle. 2. Demonstrate: The medians of a triangle meet in a point which is two-thirds of the distance from each vertex to the middle of the opposite side. 3 Demonstrate: The tangents to a circle drawn from an exterior point are equal and make equal angles with the line joining the point to the centre. 4. Find the area of an equilateral triangle in terms of its side. Denote the side by a and the area by S. 5. The chord of an arc is 24 inches and the height of the arc 9 inches. Find the diameter of the circle. 126 SEPTEMBER, 1909 9 to 10.30 A.M. (NOTE. Give reasons in full for any statement made in the course of a demonstration.) 1. Define: (a) A median of a triangle. (e) A tangent to a circle. 2. Demonstrate: The areas of two similar segments of circles are to each other as the squares of their radii. 3. Compute the altitude H of a triangle in terms of its sides A, B and C. 4. Demonstrate: The square of the bisector of an angle of a triangle is equal to the product of the sides of this angle diminished by the product of the segments determined by the bisector upon the third side of the triangle. 5. The chord of half an arc of a circle is 12 feet and the radius of the circle is 18 feet. Find the height of the arc. JUNE, 1910 (Time allowed, one hour and a half.) (NOTE. Give reasons in full for any statement made in the course of a demonstration.) 1. Define: (a) A trapezoid. (b) A trapezium. (c) The median of a trapezoid. (d) A common interior tangent to two circles. 2. Demonstrate: If a figure is symmetrical with respect to two axes perpendicular to each other, it is symmetrical with respect to their intersection as a centre. 3. Demonstrate: In the same circle or equal circles equal chords are equally distant from the centre; and conversely if the chords are equally distant from the centre they are equal. 4. Demonstrate: The bisector of an exterior angle of a triangle meets the opposite side produced at a point the distances of which from the extremities of this side are proportional to the other two sides. 5. The radii of two circles are 8 inches and 3 inches, and the distance between centres is 15 inches. Find the lengths of their common tangents. (a) An isosceles trapezoid. (b) A common exterior tangent to two circles. (c) An apothem of a polygon. (d) A re-entrant angle of a polygon. (e) A segment of a circle. 2. Demonstrate: An angle formed by a tangent and a chord is measured by one-half the intercepted arc. 3. Demonstrate: If a straight line divide two sides of a triangle proportionally, it is parallel to the third side. 4. Construct: A parallelogram equivalent to a given square, and having the difference of its base and altitude equal to a given line. 5. From the end of a tangent 20 inches long a secant is drawn through the centre of the circle. If the exterior segment of this secant is 8 inches, find the radius of the circle. EXAMINATIONS IN PLANE GEOMETRY FOR ADMISSION TO PRINCETON JUNE, 1903 Omit one question. 1. The bisectors of the angles of any triangle meet in a point. 2. If through a point of a circle a tangent and a chord be drawn, the angle between these lines is measured by half the intercepted arc. 3. Two triangles having an angle of the one equal to an angle of the other are to each other as the products of the sides including the equal angles. 4. The sides of a triangle are 5, 5, and 8 meters respectively; find the radius of a circle whose area is equal to that of the triangle. 5. The side of a rhombus is 52 meters and one of its diagonals is 96 meters; find its area. 6. In a quadrilateral the diagonals are equal and also one pair of opposite sides is equal. Prove that two of the triangles into which the quadrilateral is divided by the diagonals are isosceles. 7. The common points A and B of two circles are joined to a point P on one of the circles and the lines PA and PB cut the other circles again in C and D respectively. Prove that CD is parallel to the tangent at P. 8. The bisectors of the exterior angles at B and C of a triangle ABC meet in a point D. Show that twice the angle BDC is the supplement of the angle A. JUNE, 1904 Omit one question. 1. The sum of two lines drawn from a point to the extremities of a straight line is greater than the sum of two other lines similarly drawn but included by them. 2. The perpendicular erected at the extremity of a radius of a circle is tangent to the circle. 3. In any obtuse-angled triangle the square of the side opposite the obtuse angle is equal to the sum of the squares of the other two sides plus twice the product of one of these sides and the projection of the other side upon it. 4. If the diagonal AC of a quadrilateral ABCD divides it into two equivalent triangles, AC bisects BD. 5. If a hexagon has its opposite sides equal and parallel, the three lines joining the opposite angles meet in a point. 6. In the triangle ABC, D is the foot of the perpendicular from A upon the base BC, and E and F are the middle points of the sides AB and AC respectively; prove that the angle EDF is equal to the angle EAF and that the area of the quadrilateral EDFA is equal to one-half that of the triangle ABC. 7. Find the radius of the circle whose area is equal to that of a regular hexagon whose perimeter is 36. 8. The sides of a triangle are 3, 6, 7; find the perimeter of a similar triangle whose area is three times as great as that of the given triangle, JUNE, 1905 Omit one question. 1. If two circles touch each other internally, the straight line which joins their centres being produced will pass through the point of contact. |