EXERCISES ON BOOK VI.

THEOREMS.

1. IF a straight line AB is parallel to a plane MN, any plane perpendicular to the line AB is perpendicular to the plane MN. (v. Proposition VI., Exercise.)

2. If a plane is passed through one of the diagonals of a parallelogram, the perpendiculars to this plane from the extremities of the other diagonal are equal.

3. If the intersections of a number of planes are parallel, all the perpendiculars to these planes, drawn from a common point in space, lie in one plane.

Suggestion. Through the common point pass a plane perpendicular to one of the intersections. (v. Proposition XV., Corollary II.)

4. If the projections of a number of points on a plane are in a straight line, these points are in one plane.

5. If each of the projections of a line AB upon two intersecting planes is a straight line, the line AB is a straight line.

6. Two straight lines not in the same plane being given: 1st, a common perpendicular to the two lines can be drawn; 2d, the common perpendicular is the shortest distance between the two lines.

Suggestion. Let AB and CD be the two given lines. Pass through AB a plane MN parallel to CD, and through AB and CD pass planes perpendicular to MN. Their intersection Cc is the required common perpendicular. CD and cd are parallel, by 18, Exercise.

2d. Any other line EF joining

M

с

E

D

H

d

G

N

AB and CD is greater than EH, the perpendicular from E to cả (Proposition XV.), and therefore greater than Cc.

M

7. If two straight lines are intersected by three parallel planes, their corresponding segments are proportional. (v. Proposition VIII.)

8. A plane passed through the middle point of the common perpendicular to two straight lines in space, and parallel to both these lines, bisects every straight line joining a point of one of these lines to a point of the other. (v. Exercise 7.)

9. In any triedral angle, the three planes bisecting the three diedral angles intersect in the same straight line. (v. 40, Exercise.)

10. In any triedral angle, the three planes passed through the edges and the bisectors of the opposite face angles respectively intersect in the same straight line.

S

B

Suggestion. Lay off equal distances SA, SB, SC, on the three edges, and pass a plane through A, B, C. The intersections of the three planes in question with ABC are the medial lines of ABC, and have a common intersection, and the line joining this common intersection with S lies in the three planes. 11. In any triedral angle, the three planes passed through the bisectors of the face angles, and perpendicular to these faces respectively, intersect in the same straight line.

Suggestion. Use the same construction as in Exercise 10. Then the intersections of the three planes with ABC are perpendicular to the sides of ABC at their middle points, and have a common intersection.

12. In any triedral angle, the three planes passed through the edges, perpendicular to the opposite faces respectively, intersect in the same straight line.

Suggestion. At any point A of one of the edges, draw a plane ABC perpendicular to the edge SA. The intersections of the three planes with ABC are the perpendiculars from the vertices of ABC, upon the oppo

8

B

site sides, and have a common intersection. (v. Proposition XVI.)

LOCI.

13. Find the locus of the points in space which are equally distant from two given points.

14. Locus of the points which are equally distant from two given straight lines in the same plane.

15. Locus of the points which are equally distant from three given points.

16. Locus of the points which are equally distant from three given planes. (v. 40, Exercise.)

17. Locus of the points which are equally distant from three given straight lines in the same plane.

18. Locus of the points which are equally distant from the three edges of a triedral angle (Exercise 11).

19. Locus of the points in a given plane which are equally distant from two given points out of the plane.

20. Locus of the points which are equally distant from two given planes, and at the same time equally distant from two given points.

PROBLEMS.

In the solution of problems in space, we assume,-1st, that a plane can be drawn passing through three given points (or two intersecting straight lines) and its intersections with given straight lines or planes determined; and, 2d, that a perpendicular to a given plane can be drawn at a given point in the plane, or from a given point without it. The actual graphic construction of the solutions belongs to Descriptive Geometry.

21. Through a given straight line, to pass a plane perpendicular to a given plane. (v. Proposition XVII.)

22. Through a given point, to pass a plane perpendicular to a given straight line.

Suggestion. If the given point is in the given line, pass two planes through the given line, and draw in each of them, through the given point, a line perpendicular to the given line. The plane determined by these lines is the perpendicular plane required. (v. Proposition IV.)

If the given point is not in the given line, pass a plane through it and the given line, and in this plane, through the given point, draw a line parallel to the given line. A plane through the given point, perpendicular to this second line, is the plane required. (v. Proposition XI.)

23. Through a given point, to pass a plane parallel to a given plane. (v. Proposition IX., Corollary.)

24. To determine that point in a given straight line which is equidistant from two given points not in the same plane with the given line. (v. Exercise 13.)

25. To find a point in a plane which shall be equidistant from three given points in space.

26. Through a given point in space, to draw a straight line which shall cut two given straight lines not in the same plane.

Suggestion. Pass a plane through the given point and through one of the given lines; the line through the given point and the point where the plane cuts the second given line is the solution required.

27. Through a given point, to draw a straight line which shall meet a given straight line and the circumference of a given circle not in the same plane. (Two solutions in general.)

28. In a given plane and through a given point of the plane, to draw a straight line which shall be perpendicular to a given line in space.

Suggestion. Draw a plane through the given point and perpendicular to the given line. Its intersection with the given plane is the solution required.

29. Through a given point A in a plane, to draw a straight line AT in that plane, which shall be at a given distance PT from a given point P without the plane.

Suggestion. Drop a perpendicular from P to the plane, and with the foot of this perpendicular as a centre, and with a radius equal to a side of a right triangle whose hypotenuse is PT, and whose other side is the length of the perpendicular, describe a circumference in the plane. A tangent from A to this circumference is the solution required. (v. 12, Exercise 2.)

30. Through a given point A, to draw to a given plane M a straight line which shall be parallel to a given plane N and of a given length.

BOOK VII.

POLYEDRONS.

1. DEFINITION. A polyedron is a geometrical solid bounded by planes.

The bounding planes, by their mutual intersections, limit each other, and determine the faces (which are polygons), the edges, and the vertices of the polyedron. A diagonal of a polyedron is a straight line joining any two of its vertices not in the same face.

The least number of planes that can form a polyedral angle is three; but the space within the angle is indefinite in extent, and it requires a fourth plane to enclose a finite portion of space, or to form a solid; hence the least number of planes that can form a polyedron is four.

2. Definition. A polyedron of four faces is called a tetraedron; one of six faces, a hexaedron; one of eight faces, an octaedron; one of twelve faces, a dodecaedron; one of twenty faces, an icosaedron.

3. Definition. A polyedron is convex when the section formed by any plane intersecting it is a convex polygon.

All the polyedrons treated of in this work will be understood to be convex.

4. Definition. The volume of any polyedron is the numer ical measure of its magnitude, referred to some other polyedron as the unit. The polyedron adopted as the unit is called the unit of volume.

To measure the volume of a polyedron is, then, to find its ratio to the unit of volume.