96. The areas of two concentric are to each other as 5 to 8. Find the radii of the two ©, if the area of that part of the ring which is contained between two radii making the angle 45° is 300 square feet. 97. If two tangents, including an of 60° and drawn from the same point without a O, with two radii drawn to their points of contact, inclose an area of 162√3ea, find the length of these tangents and the area of the sector formed by these two radii and their arc. 98. Find the area of the segments of the O in the preceding problem made by a chord perpendicular to its radius at its middle point. 99. If a track, having two parallel sides and two semicircular ends, each equal to one of the parallel sides, measures exactly a mile at the curb, what distance does a horse cover running ten feet from the curb? How many acres within the circuit he makes? Ha 100. Three, each tangent to the other two, inclose with their convex arcs 1a of ground. How far is it from the centres of these to the middle point of this piece of ground? NUMERICAL PROBLEMS, EXERCISES, PROPOSI TIONS, AND OTHER QUESTIONS SELECTED FROM THE ENTRANCE EXAMINATION PAPERS OF A NUMBER OF THE LEADING COLLEGES AND SCIENTIFIC SCHOOLS. 1. From any point in the base of an isosceles triangle perpendiculars are drawn to the sides; prove their sum to be equal to the perpendicular drawn from either basal vertex to the opposite side.-Boston University. 2. The angle at the vertex A of an isosceles triangle ABC is equal to twice the sum of the equal angles B and C. If CD is drawn perpendicular to BC, meeting A B produced at D, prove that the triangle A CD is equilateral.- Wesleyan University. 3. If from one of the vertices (A) of a triangle (A B C) a distance (A D) equal to the shorter one of the two sides (A B and A C) meeting in A be cut off on the longer one (AB), prove that DC B = [ /A C B − ZA B C]. — U. of Cal. 4. Show that the angle included between the internal bisector of one base angle of a triangle and the external bisector of the other base angle is equal to half the vertical angle of the triangle.-Harvard. 5. If ABC be an equilateral triangle, and if BD, CD bisect the angles B, C, the lines DE, DF parallel to A B, A C, divide BC into three equal parts.-Cornell. 6. What is a polygon? Prove that the sum of the interior angles of an n-gon is n 2 straight angles.-Dartmouth. 7. AD and BC are the parallel sides of a trapezoid ABCD, whose diagonals intersect at E. If F is the middle point of BC, prove that EF produced bisects A D.-Mass. Inst. Tech. 8. If perpendiculars be drawn from the angles at the base of an isosceles triangle to the opposite sides, the line from the vertex to the intersection of the perpendiculars bisects the angle at the vertex and the angle between the perpendiculars. Prove.-Boston University. 9. Prove that a parallelogram is formed by joining the midpoints of the (adjacent) sides of any quadrilateral. Hint, draw the diagonals of the quadrilateral.—Bowdoin. 10. In any triangle A B C, if AD is drawn perpendicular to BC, and AE bisecting the angle BA C, the angle DAE is equal to one-half the difference of the angles B and C.-Cornell. 11. Show that in any right-angled triangle the distance from the vertex of the right angle to the middle point of the hypotenuse is equal to one-half the hypotenuse. School of Mines. 12. If D is the middle point of the side BC of the triangle A B C, and BE and CF are the perpendiculars from B and C to AD, prove that B E = C F.- Wesleyan University. 13. If in a right-angled triangle one of the acute angles is one-third of a right angle, the opposite side is one-half the hypotenuse.-U. of Cal. 14. Prove that the diagonals and the line which joins the middle points of the parallel sides of a trapezoid meet in a point. Harvard. 15. How many degrees in one angle of an equiangular docedagon ?-Dartmouth. 16. If the opposite sides of a pentagon be produced to intersect, prove that the sum of the angles at the vertices of the triangles thus formed is equal to two right angles.Cornell. 17. The interior angle of a regular polygon exceeds the exterior angle by 120°. How many sides has the polygon? -Mass. Inst. Tech. 18. If one diagonal of a quadrilateral bisects both angles whose vertices it connects, then the two diagonals of the quadrilateral are mutually perpendicular. Prove.-—Boston University. 19. In a given polygon, the sum of the interior angles is equal to four times the sum of the exterior. How many sides has the given polygon ?-Wesleyan University. 20. What is the greatest number of re-entrant angles a polygon may have compared to the number of its sides? What is the value of the re-entrant angles of a pentagon in terms of the interior angles not adjacent ?-Cornell. 21. Show what the sum of the opposite angles of a quadrilateral inscribed in a circle is equal to.-Columbia. 22. When and why may an arc be used as the measure of an angle? The vertex of an angle of 60° is outside a circle and its sides are secants; what is the relation between the intercepted arcs ?-Dartmouth. 23. Show that two angles at the centres of unequal circles are to each other as their intercepted arcs divided by the radii.-U. of Cal. 24. Prove that in any quadrilateral circumscribed about a circle the sum of two opposite sides is equal to the sum of the other two opposite sides. -Harvard. 25. Construct a common tangent to two circles.-Boston University. 26. Three consecutive sides of a quadrilateral inscribed in a circle subtend arcs of 82°, 99°, and 67° respectively. Find each angle of the quadrilateral in degrees, and the angle between its diagonals.— Yale. 27. If A C and B C are tangents to a circle whose centre is O, from a point C without the circle, prove that the centre of the circle which passes through O, A, and B, bisects O C.-Mass. Inst. Tech. 28. Fix the position of a given circle that touches two intersecting lines.- Vanderbilt University. 29. Through a given point in the circumference of a circle chords are drawn. Find the locus of their middle points. Cornell. 30. Give contractions for the inscribed, escribed, and circumscribed circles of any triangle.-Sheffield S. S. 31. Construct a circle that shall pass through two given points and shall cut from a given circle an arc of given length.- Vassar. 32. Prove that the circumference of a circle may be passed through the vertices of a quadrilateral provided two of its opposite angles are supplementary.-Boston University. 33. A and B are two fixed points on the circumference of a circle, and PQ is any diameter. What is the locus of the intersection of PA and QB ?-Harvard. 34. The length of the straight line joining the middle |