4. Take a straight line AB, a line segment s, and a point P. Find a point X on AB at a distance s from P. Under what conditions is the problem impossible? How many solutions can it have?

5. Take two points P and Q, and two line segments r and s. Find a point X at a distance r from P and a distance s from Q. Is this problem ever impossible? How many points X may be found?

6. Take two points P and Q, and a line segment r. distance r from P and Q. Discussion.

Find a point X at a

7. With a given radius r describe a circle passing through two given points P and Q. Discussion.

8. The following figures represent the geometric bases of three types of arches used in architecture called, respectively, the equilateral arch, the segmental arch, and the Moorish arch.

D

Construct Fig. (a), given the span AB = 1 inches.

Construct Fig. (b), given the span AB = 11⁄2 inches, and the radius CA= 1 inch.

Construct Fig. (c), given the span AB = 11⁄2 inches,

MOTION

CONGRUENT FIGURES

28. Motion. When we defined a circle we imagined a line segment to move in a certain way (§ 18). Also in discussing the equality of line segments the idea of motion was implied (§ 9, see also § 11). The fact that a figure may be thought of as moving from one position to another is of such fundamental importance in the study of geometry, that it is desirable that we gain a clear idea of the motions that are possible. We will readily agree to the following:

29. FUNDAMENTAL PROPOSITIONS. I. Any figure in a plane may be imagined to slide along the plane in such a way that every point in the figure moves along a straight line.

30. II. Any point C of the plane of a figure may be kept fixed and the figure imagined to rotate in the plane about C as a center. Every point of the figure (except C) will then describe a circle with center C.

31. III. Any two points S and T of the plane of a figure may be kept fixed and the plane imagined to rotate about the line ST as an axis, until it coincides again with its original position, but with its sides reversed.

We shall describe this type of motion by saying that the figure has been turned over about the axis ST. During such a motion every point of the axis remains fixed.

32. Congruence of figures. On the idea of motion depends a fundamental relation which may exist between two figures and which is defined as follows: Two figures are said to be identically

equal or congruent, if by a suitable motion the figures can be placed so as to coincide throughout.

33. If two figures in the same plane are congruent, they may be made to coincide by the successive application of one or more of the types of motion I, II, III, described in §§ 29, 30, 31. The following propositions will illustrate the truth of this statement.

34. Two circles with equal radii are congruent. Let the circles be in the same plane with centers C and C. Imagine the first circle to slide so that its center C moves along the straight line CC until it coincides with C. Then, since their radii are equal, the circles will coincide throughout (§ 22). Here a slide was sufficient to show the figures congruent.

35. If we wish to bring two equal line segments, AB and CD, into coincidence, we need only imagine AB to slide so that A moves along the line AC until A coincides with C, and then

imagine AB to rotate about C until B coincides with D. Here

a slide followed by a rotation suffices.

Cases where the third type of motion is required to establish congruence will appear almost immediately.

SYMMETRY WITH RESPECT TO A STRAIGHT LINE

36. A symmetric figure. Imagine Fig. 9 to rotate about the straight line MN as an axis until it comes again into the plane of its original position. Let Fig. 10 be a duplicate of Fig. 9 together with the latter after the rotation. Figure 10 is said to be symmetric with respect to the axis MN. It should be observed that Fig. 10 has the property that, if it is turned over on the axis MN, it will coincide throughout with its original position.

37. Symmetry with respect to a straight line. In general, a figure is said to be symmetric with respect to a straight line as an axis of symmetry, if, after the figure has been turned over on the axis, it coincides throughout with its original position.

38. Corresponding parts of a symmetric figure. Any two parts of a symmetric figure are said to be corresponding which simply change places when the figure is turned over on the axis of symmetry. Either one of any two corresponding parts is called the reflexion of the other with respect to the axis of symmetry.

39. If a figure is symmetric with respect to a straight line, corresponding parts are congruent (§ 32).

40. If a figure is symmetric with respect to a straight line, the lines joining corresponding points are symmetric with respect to the axis.

For, if A, A' (Fig. 10) are two corresponding points, the latter will, when the figure is turned over on the axis, simply change places. The line AA' will therefore coincide with its original position after it has been turned over on the axis (§ 3).

41. Mid-point of a line segment. A point M of a line segment AB is called the mid-point of the segment, if the segments AM and MB are equal. The point M is then said to bisect the segment. Also any straight line through M is said to bisect the segment.

The following proposition will be readily admitted.

42. FUNDAMENTAL PROPOSITION. The line segment joining two points on opposite sides of a straight line has a point in common with the line.

43. If a figure is symmetric with respect to a straight line, the line segments joining corresponding points are bisected by the axis of symmetry. Thus in Fig. 10 the segment AA' is bisected by the axis MN (§ 39).

44. A circle is symmetric with respect to any straight line passing through its center.

For, if the circle be turned over on any straight line through the center, the latter remains fixed. The circle, after being turned over, then coincides with its original position, since it has the same center and the same radius (§ 22).

45 As a very useful consequence, we have the following:

The figure formed by any two circles is symmetric with respect to the line joining the centers A of the two circles.

46. In connection with this proposition it should be noted that if two circles intersect in two points A, B, these two points are corresponding points in the symmetry with respect to the line joining the centers. The line AB is then

N

D

M

FIG. 11.

symmetric with respect to this axis, and the segment AB will be bisected by the axis. This fact will be made the basis of the following important construction.