CHAPTER XIII.

EQUALITY AND SIMILARITY OF POLYGONS.

315. We are able to investigate the conditions of equality of polygons only when they are regular.

Two regular polygons are equal when the side of the one is equal to the side of the other and the interior angles are the same.

316. It is evident that they will coincide when superposed, for if we make the side A'B' of the one coincide with the side A B of the other, which is equal to it, then B'C' must take the direction B C, since the angle B' is equal to the angle B, and the point C' will fall on C since the sides are equal, etc.

Two regular polygons which have the same number of sides are equal when they are inscribed in equal circles.

317. For we can superpose the circumferences, and make two angles coincide, and then all the other angles must necessarily coincide.

To construct a regular polygon equal to a given polygon.

318. Measure the radius of the circle described about the given polygon, and draw another circle of the same radius. Take upon its circumference a chord equal to the length of the side of the given polygon, and repeat the operation until all the angular points of the polygon to be described are marked on the circumference. When the side only of the polygon is given, calculate the angle at the centre of the polygon, and upon the given side as base, make an isosceles triangle having this angle at the vertex. The apex of this angle will be the centre of the

circle described about the required polygon. In the case of some polygons simpler constructions may be given, the demonstrations of which we may now leave to the student.

319. Upon a given straight line, A B, to construt a hexagon (fig. 333).

Describe circles with A and B as centres, and A B as radius, and a third circle from O, their point of intersection, with the same radius, meeting the other circles in C and D. These will be two other angular points of the hexagon required. Two new arcs described from C and D will give the remaining points E and F. 320. Upon a given straight line, A B, to describe an octagon (fig. 334).

B

Сс

B

Fig. 333.

Fig. 334.

At the middle point of the given side A B erect a perpendicular.

Fig. 335.

B

Take CD equal to CA, and DO equal to D A. O will be the centre of the circumscribed circle.

321. Upon a given straight line, AB, to describe a decagon (fig. 335). At the point B erect a perpendicular C B, equal to half A B, and describe a circle from the centre C, with radius CB; draw A C; then A D, or its equal A O, will be the

radius of the circumscribed circle.

322. Upon a given straight line, A B, to describe a dodecagon (fig. 336).

At the middle of A P, the given side, erect a perpendicular; from A as centre, with radius A B, describe a circle cutting the perpendicular in the point D; from D as centre, with the same radius, describe a circle cutting the perpendicular in the point O; then O is the centre of the circumscribed circle.

To construct a polygon equal to a given polygon. 323. Let ABCDEFGH be the given polygon (fig. 337). Divide it by diagonals into the triangles AFD,

FDE, ADC, A BC, A HF, and FGH, and construct triangles equal to these component triangles and in the same order. The polygon thus formed will be equal to the given polygon; for their sides will be equal each to each, and so also will be the angles contained by the equal sides.

The following is another method :—

324. Draw a diagonal, A F, and from every apex let fall a perpendicular upon it (fig. 338.) Now draw a line

equal to this diagonal. Mark upon it the positions of

the points b, c, d, e, etc., and upon the perpendiculars erected at these points take distances respectively equal to those of the first figure. As before, a polygon is formed equal to the given polygon.

M

D

Fig 339.

E

N

F

325. Instead of a diagonal, we may draw at

P n

pleasure two, three M
or four lines cutting
one another, and pro-
ceed in a similar
manner (fig. 339.)
Again, we may
project the angular
points of the polygon
upon the sides of a
rectangle, for this is
figure easy of construction
(fig. 340.) Another rect-
angle may be then made.
equal to the first, and the
positions of the different
points marked upon its
sides.

a

326. By drawing through the angles of the given polygon, equal and parallel straight lines, and joining their extremities, we shall obtain a polygon equal to the first. This method may be applied to the outline of any figure whatever (fig. 341).

To copy charts, models, embroidery, etc., on a surface divided into squares.

327. Instead of using a single rectangle for reference, the whole figure may be covered with rectangles. Usually the square, which is the simplest of rectangles, is chosen (figs. 342, 343).

If these squares are made so small that, without measuring, we can mark the distances of a point in the figure from the sides of the square in which it is placed, the copy may be made with great rapidity. This method is generally used when the figure contains many very small lines, and particularly when they are so small as to form a curve. It is useful for copying a map or drawing. In designs for wool-work, the squares mark the place for the point of the needle, so that it is very easy to reproduce the figure upon canvas, the tissue of which is a system of squares (fig. 344.)