CHAPTER III.

PROBLEMS RELATING TO LINES.

I. To bisect a line A B (either straight or curved).From A, with any distance greater than the half of A B,

B

D

B

F

Fig. 1.

describe the arc C D E, and from B, with the same distance, the arc C F E. Join C E. It bisects A B in G.

Note. This may be conveniently and quickly done by two or three trials with the dividers.

II. To draw lines at right angles from any point C. 1st Method (fig. 2).-From C, with any radius, describe the arc D E. With the same radius cut that arc in the points a and b, and from a and b describe arcs cutting in F. Join C F, the line required.

2nd Method (fig. 3).-From C, with any distance, cut A B in the points a and b. From a and b, with distance a b, describe arcs cutting in D. Join D C, the line required.

3rd Method (fig. 4).-From any point D, above A B and between A and B, with radius D C, describe a circle. Set off the radius on the circle from E in the points 1, 2, 3. Join 3 C, the line required.

4th. Builder's Method (fig. 5).-Builders set out their walls at right angles by taking three rods; one is made 3 yards long, the other 4 yards, and the hypotenuse 5 yards long. The angle C is then a right angle.

5th Method (fig. 6).— Lines may be quickly drawn at right angles by the set square.

In Fig. 6, A B is the line, C the point, and C D E the set square.

Sel
Square

E

Fig. 6.

III. To drop a perpendicular from a point C on to

a line A B.

Fig.7.

Fig. 8.

2nd Method (fig. 8). --Draw any line CD, meeting A B in D. Bisect CD in E, and from E as centre, with E Cas radius, describe a circle cutting A B in F, and join C F.

Note.-CD may be bisected by the dividers.

3rd Method (fig. 9).-Perpendicular lines may be quickly drawn by means of two set squares, or by a set square and ruler. Fig. 9 will show the method of proceeding to drop a perpendicular from C on to the line A B.

Sel Square

Set Square

Fig. 9.

Ruler

The dotted lines show the perpendiculars required.
IV. To draw a line parallel to a given line, A B.

1st Method (fig. 10).-When the line is to be at any given distance from A B (e.g. 3 inch). With radius inch from

any points c and d in A B describe arcs. Draw the line C D to touch the tops of the arcs. This is the required

line.

B

D

Fig. 11.

2nd Method (fig. 11).—The line to pass through a given point C. From C with any radius describe the arc De, and from D with radius D C the arc Cf. Set off De equal to Cƒ and draw C e. The line Ce is parallel to AB. 3rd Method. By set squares, or ruler and set square. Fig. 12 will show the method of drawing the lines parallel to A B.

Set one edge of the set square to the line A B. by sliding set square 1 on set square 2 or on the ruler,

Then

the lines C D and E F will be parallel to A B.

CHAPTER IV.

PROBLEMS RELATING TO ANGLES.

I. To Construct a Scale of Chords for the Construction and Measurement of Angles.-By means of the set square or other method make

the right angle A B C, and from the centre B 90 at any distance BD describe the quadrant D E. Divide D E into 9 equal parts. Each part is then 10 degrees of magnitude. Join D E, and from the centre D with the distance D 1, D2, D 3, etc., describe arcs cutting the B straight line D E. DE is the scale of chords, and is generally marked

on scale rulers Cho., or Ch., or sometimes C. Mark the degrees on D E, as shown.

II. Construction of Angles by Scale of Chords.

from the scale, and cut this arc in the point 2. The angle 2 B A is 35°.

Join B 2.