Ex. 924. The volume of a sphere is to that of the inscribed cube as π

Ex. 925. The surface of a sphere is to the total surface of the circumscribing cylinder as 2 is to 3.

Ex. 926. The volume of a sphere is to the volume of a circumscribing cylinder as 2 is to 3.

Ex. 927. A sphere is cut by five parallel planes at the distance of 2dm, 3dm, 4dm, and 5dm from each other, respectively. What are the relative areas of the zones included between the planes ?

Ex. 928. The sides opposite the equal angles of a birectangular spherical triangle are quadrants.

Ex. 929. The slant height of a cone of revolution is equal to the diameter of its base. What is the ratio of its total area to that of the inscribed sphere?

Ex. 930. The smallest circle whose plane passes through a given point within a sphere is that one whose plane is perpendicular to the radius through the given point.

Ex. 931. The intersection of the surfaces of two spheres is the circumference of a circle whose plane is at right angles to the line joining the centers of the spheres, and whose center is on that line.

Ex. 932. What is the area of the circle of intersection of two spheres whose radii are respectively 5dm and 8dm, if their centers are 10dm apart?

Ex. 933. What is the weight of an iron ball, the area of whose surface is 2sq m, the specific gravity of iron being 7.5?

Ex. 934. If the exterior diameter of a spherical shell is 12 in., what should be the thickness of its wall in order that it may contain 696.9 cu. in. ?

Ex. 935. What is the weight of a hollow iron shell whose wall is 2 in. thick, if it will hold 31 pounds of water, the specific gravity of iron being 7.5?

Ex. 936. An equilateral triangle revolves about its altitude. Compare the volumes of the solids generated by the triangle, the inscribed circle, and the circumscribed circle.

Ex. 937. From a sphere whose surface is 69 sq. ft. a segment of one base is cut, which has an altitude of 3 ft. What is the convex surface of the segment ?

Ex. 938. What is the radius of a sphere inscribed in a regular tetrahedron whose entire area is 16 sq. ft. ?

Ex. 939. What is the area of the surface of a sphere inscribed in a regular tetrahedron whose edge is 6 in. ?

Ex. 940. How much of the surface of the earth could a man see, if he were at the distance of a diameter above it?

Ex. 941. How far from the surface of the earth must a man be in order that he may see one fifth of it?

Ex. 942. All arcs of great circles drawn through the pole of a given great circle are perpendicular to its circumference.

Ex. 943. The sum of the squares of three chords perpendicular to each other at any point in the surface of a sphere is equal to the square of the diameter.

Ex. 944. If a zone of one base is a mean proportional between the remaining surface of the sphere and its total surface, how far is the base of the zone from the center of the sphere?

Ex. 945. If any number of lines in space meet in a point, the feet of the perpendiculars drawn to these lines from another point lie in the surface of a sphere.

PROBLEMS OF CONSTRUCTION

Ex. 946. Bisect an arc of a great circle.

Ex. 947. Bisect a spherical angle.

Ex. 948. At a given point in a given arc of a great circle construct a spherical angle equal to a given spherical angle.

Ex. 949. Construct a spherical triangle, the poles of the respective sides being given.

Ex. 950. Construct a spherical triangle, having given two sides and the included angle.

Ex. 951. Construct a spherical triangle, having given a side and two adjacent angles.

Ex. 952. Construct a spherical triangle, having given the three sides. Ex. 953. Construct a spherical triangle, having given the three angles. Ex. 954. Draw an arc of a great circle perpendicular to a given spherical arc from a point without.

Ex. 955. Draw an arc of a great circle perpendicular to a given spherical arc at a point in it.

Ex. 956. Pass a plane tangent to a sphere at a given point on the surface of the sphere.

Ex. 957. Pass a plane tangent to a sphere through a given straight line vithout the sphere.

Ex. 958. Cut a given sphere by a plane passing through a given straight line so that the section shall have a given radius.

Ex. 959. Through a given point on a sphere draw a great circle tangent to a given small circle.

Ex. 960. Through a given point on a sphere draw a great circle tangent to two equal small circles whose planes are parallel.

Ex. 961. Describe a circle to pass through three given points on the surface of a sphere.

Ex. 962. Circumscribe a circle about a given spherical triangle.

EXERCISES FOR REVIEW

Ex. 1. The perpendicular erected at the middle point of one side of a triangle meets the longer of the other two sides.

Ex. 2. Of the bisectors of two unequal angles of a triangle, produced to the point of intersection, the bisector of the smaller angle is the longer.

Ex. 3. The straight lines which join the middle points of the opposite sides of any quadrilateral bisect each other.

Ex. 4. If a line is drawn from the middle point of one base of a trapezoid to pass through the intersection of the diagonals, it will bisect the other base. Ex. 5. If the opposite sides of a pentagon are produced to intersect, the sum of the angles at the vertices of the triangles thus formed is equal to two right angles.

Ex. 6. The sum of the four lines drawn from the vertices of any quadrilateral to any point except the intersection of the diagonals is greater than the sum of the diagonals.

Ex. 7. If the internal bisector of one base angle of a triangle and the external bisector of the other base angle are produced until they meet, the angle included between them is equal to half the vertical angle of the triangle.

Ex. 8. The angle contained by the bisectors of two exterior angles of any triangle is equal to half the sum of the two adjacent interior angles.

Ex. 9. If each of two angles of a quadrilateral is a right angle, the bisectors of the other angles are either perpendicular or parallel to each other.

Ex. 10. If the side CB of the triangle ABC is greater than the side CA, and CA is produced to D and CB to E, making AD and BE equal, AE is greater than DB.

Ex. 11. In the triangle ABC a straight line AD is drawn perpendicular to the straight line BD which bisects angle B. Prove that a straight line through D parallel to BC bisects AC.

Ex. 12. If one side of a triangle is greater than the other, any line from the vertex of the included angle to the base is less than the longer side.

Ex. 13. Lines drawn from one vertex of a parallelogram to the middle points of the opposite sides trisect a diagonal.

Ex. 14. No two straight lines drawn from two vertices of a triangle and terminated by the opposite sides can bisect each other.

Ex. 15. The base of a triangle whose sides are unequal is divided into two parts by a straight line bisecting the vertical angle. Prove that the greater part is adjacent to the greater side.

Ex. 16. If two exterior angles of a triangle are bisected, and from the point of intersection of the bisectors a straight line is drawn to the vertex of the third angle, this line bisects that angle.

Ex. 17. ABC is a triangle; D is the middle point of BC, and E of AD; BE produced meets AC in F. Prove that AC is trisected in F.

Ex. 18. In the triangle ABC the sides AB, BC, and CA are trisected at the consecutive points D and E, F and G, and H and K respectively. Prove that the lines EF, GH, and KD, when produced, form a triangle equal to ABC.

Ex. 19. If one of the equal sides of an isosceles triangle is produced below the base to a certain length, if an equal length is cut off above the base from the other equal side, and if the two points are joined by a straight line, this line is bisected by the base.

Ex. 20. ABC is a triangle, and BE and CF are drawn perpendicular to AG, a line through A; D is the middle point of BC. Show that FD equals ED.

Ex. 21. The angle contained by the bisectors of the base angles of any triangle is equal to the vertical angle of the triangle plus half the sum of the base angles.

Ex. 22. The bisectors of two angles of an equilateral triangle intersect, and from their point of intersection lines are drawn parallel to any two sides. Prove that these lines trisect the third side.

Ex. 23. The opposite sides of a regular hexagon are parallel.

Ex. 24. If in a quadrilateral the diagonals are equal and two sides are parallel, the other sides are equal.

Ex. 25. If from any point in the base of an isosceles triangle perpendiculars are drawn to the equal sides, their sum is equal to the perpendicular drawn from either extremity of the base to the opposite side.

Ex. 26. The sum of the perpendiculars from any point within an equilateral triangle to its sides is equal to the altitude.

Ex. 27. If from the vertex of any triangle two lines are drawn, one of which bisects the angle at the vertex and the other is perpendicular to the base, the angle between these lines is half the difference of the angles at the base of the triangle.

Ex. 28. In any triangle, the sides of the vertical angle being unequal, the median drawn from the vertical angle lies between the bisector of that angle and the longer side.

Ex. 29. In any triangle, the sides of the vertical angle being unequal, the bisector of that angle lies between the median and the perpendicular drawn from the vertex to the base.

Ex. 30. Lines are drawn through the extremities of the base of an isosceles triangle, making angles with it, on the side opposite the vertex, each equal to one third of a base angle of the triangle, and meeting the sides produced. Prove that the three triangles thus formed are isosceles.

Ex. 31. If two circumferences are tangent internally and the radius of the larger is the diameter of the smaller, any chord of the larger drawn from the point of contact is bisected by the circumference of the smaller.

Ex. 32. If perpendiculars are drawn to any chord at its extremities and produced to intersect a diameter of the circle, the points of intersection are equally distant from the center.

Ex. 33. If perpendiculars are drawn from the ends of a diameter of a circle upon any secant, their feet are equally distant from the points in which the secant intersects the circumference.

Ex. 34. Given an arc of a circumference, the chord subtended by it, and the tangent at one extremity. Prove that the perpendiculars dropped from the middle point of the arc upon the tangent and chord, respectively, are equal.

Ex. 35. The bisectors of the angles contained by the opposite sides (produced) of an inscribed quadrilateral intersect at right angles.

Ex. 36. If two opposite sides of an inscribed quadrilateral are equal, the other two sides are parallel.

Ex. 37. In a given square, inscribe an equilateral triangle having its vertex in the middle of a side of the square.

Ex. 38. Find, in a side of a triangle, a point from which straight lines, drawn parallel to the other sides of the triangle and terminated by them, are equal.

Ex. 39. Construct a triangle, having given the base, one of the angles at the base, and the sum of the other two sides.

Ex. 40. Construct a triangle, having given the base, one of the angles at the base, and the difference of the other two sides.

Ex. 41. Construct a triangle, having given the perpendicular from the vertex to the base, and the difference between each side and the adjacent segment of the base.

Ex. 42. If two circles are each tangent to two parallel lines and a transversal crossing them, the line of centers is equal to the segment of the transversal intercepted between the parallels.

Ex. 43. If through the point of contact of two circles which are tangent to each other externally any straight line is drawn terminated by the circumferences, the tangents at its extremities are parallel to each other.

Ex. 44. If two circles are tangent to each other externally and parallel diameters are drawn, the straight line joining the opposite extremities of these diameters will pass through the point of contact.

Ex. 45. Construct the three escribed * circles of a given triangle.

* A circle tangent to one side of a triangle and to the other two sides produced is called an escribed circle.