(3) Arc.--Any single portion of the circle. More special properties of the circle will be mentioned later.

53. Constructions with ruler and compass.-a. Line segment.To copy a line segment AB, figure 68 (a):

(1) With a straightedge, draw a line AK, figure 68 (b), of indefinite length, but as long as or longer than AB, figure 68 (a).

(2) Place one point of compass at A (figure 68 (b) and adjust compass so that the other point B falls on line AK.

(3) Take compass as adjusted and with one point on A, draw an arc intersecting AK at B. Then AB, figure 68 (b), is a copy of AB, figure 68 (a).

b. Angle. To copy an angle TAC, figure 69 (a):

(1) With A, figure 69 (a), as a center and with any convenient radius, strike an arc intersecting the sides of the angle at C and T.

(2) Take any point A', figure 69 (b), on an arbitrary segment L and with the same radius used in striking the arc in figure 69 (a), strike an arc intersecting line AʼL at C'.

(3) With C as a center and with radius CT (from figure 69 (a), intersect the arc previously drawn in figure 69 (b) at T'. (4) Draw line A'T'.

c. Perpendicular to a segment. To construct the perpendicular bisector of a segment AB, figure 70:

(1) With A as center and radius greater than half AB, draw an arc extending above and below the segment.

(2) Repeat (1) with B as center using same radius. This arc will intersect the first arc at P and Q.

(3) Draw line PQ intersecting AB at M. PQ is perpendicular to AB and M is the midpoint of AB.

d. Bisecting an angle.-To bisect angle BAC, figure 71.

(1) With A as center and any convenient radius, strike an arc intersecting the sides of the angle at M and N.

(2) With M and N as centers and any radius greater than half MN, describe arcs intersecting at P.

(3) Draw line AP, which bisects the angle BAC.

e. Parallel to a line through a point.-To construct a parallel to a line L though a point P, figure 72:

(1) Through P, draw any line cutting L at A.

(2) With A as a center and any radius, strike an arc cutting AP at N and L at M.

(3) With P as center and the same radius, strike an arc cutting AP at R.

(4) Take as radius the distance from N to M and with R as center, strike an arc cutting the arc previously drawn ((3) above) at S. (5) Draw line PS.

f. Perpendicular at a point.-To construct a perpendicular at a given point P in a line L, figure 73:

(1) With P as center and any radius, strike off arcs on each side of P intersecting L at M and N.

(2) With a larger radius and M and N as centers, strike off arcs intersecting at Q.

(3) Draw line PQ.

g. Perpendicular from a point.-To construct a perpendicular to a line from a point P outside the line at L, figure 74:

FIGURE 74.-Constructing perpendicular from a point.

(1) With P as center and radius large enough to intersect L, strike off an arc intersecting L at points M and N.

(2) With radius greater than one-half MN and M as center, strike off an arc on opposite side of L from P. Using same radius and N as center, strike off arc intersecting last arc at Q.

75:

(3) Draw line PQ.

h. Equilateral triangle.-To construct an equilateral triangle, figure

(1) Draw a line L of indefinite length.

(2) With any point P on L as center and any radius, strike off an arc intersecting L at M.

(3) With M as center and same radius, strike off an arc intersecting first arc at Q.

(4) Draw lines PQ and QM.

In an equilateral triangle, each angle is 60°.

i. Sum of two angles. To find the sum of two angles 1 and 2, figure 76:

(1) With A and B as centers and any radius, describe arcs cutting the sides of the angles at M, N, P, and Q, figures 76 (a) and (b).

(2) With same radius and 0—one end point of line L, figure 76 (c), as center-strike an arc intersecting L at R.

(3) With R aş center and MN as radius describe an arc intersecting the first at T.

(4) With Tas center and the distance PQ as radius, strike off an arc intersecting the first arc at S.

j. Adding angles.-Adding angles consists of copying the angles so that they have a common vertex and a common side between them. This construction, coupled with other of the basic constructions, makes it possible to construct many specific angles.

Example: Construct a 75° angle:

(1) Construct a perpendicular (90°), figure 77 (a), to a line and bisect one of the angles. This gives a 45° angle.

(2) Construct an equilateral triangle, figure 77 (b), and bisect one of the angles. This gives a 30° angle.

(3) Add the 45° and 30° angle, figure 77 (c).

k. Constructing triangle, two sides and included angle given.—To construct a triangle when two sides b and c, figure 77 (a) and the included angle 1, figure 77 (b) are given.

FIGURE 78.-Constructing triangle, two sides and included angle given.

(1) On a straight line, figure 78 (c), take AB equal to c.

(2) At A construct an angle BAC equal to 21.

(3) With A as a center and b as a radius, strike an arc intersecting AC at C.

(4) Draw line BC.

1. Constructing triangle, two angles and included side given.-To

C
(a)

44

2

(b)

A

(c)

FIGURE 79.-Constructing triangle, two angles and included side given.