38. Upon a Given Side to Draw a Regular Hexagon.

- In Fig. 107, let A B be the given side upon which a regular hexagon is to be erected. From A as center, and with A B as radius, describe the arc B C. From B as center, and with the same radius, describe the arc A C, intersecting the first arc in the point C. C will then be the center of the circle which will circumscribe the required hexagon. With C as center, and C B as radius, strike the circle, as shown. Set the dividers to the space A B and step off the circumference, as shown, obtaining the points E, G, F and D. Draw the chords A E, E G, G F, F D and D B, thus completing the required figure.

39. Upon a Given Side to Draw a Regular Hertagon.

- In Fig. 168, A B represents the given side upon which a regular heptagon is to be drawn. From B as center, and with B A as radius, strike the semicircle A E D. Produce A B to D. From A as center, and with A B as radius, strike the arc B F, cutting the semicircle in the point F. Through F draw F G perpendicular to A B, which extend in the direction of C. From D as center, and with radius G F, cut the semicircle in the point E. Draw the line E B, which is another side of the required heptagon. Bisect E B, and upon its middle point, H, erect a perpendicular, which produce until it meets the perpendicular erected upon the center of the given side A B, in the point C. Thru C is the center of the circle which will circumscribe the required heptagon, From C as center, and with C B as radius, strike the circle. Set the dividers to the distance A B and stop off the circumference, as shown, obtaining the points K, N, M and L. Draw the connecting arcs A K, KN, N M, M L and L E, thus completing the figure.

40. Upon a Given Side to Draw a Regular Octagon. - in Fig. 169, let A B represent the given side upon which a regular octagon is to be constructed. Produce A B indefinitely in the direction of D. From B as center, and with A B as radius, describe the semicircle A E D. At the point B erect a perpendicular to A B, as shown, cutting the circumference of the semicircle in the point E. Bisect the arc E D, obtaining the point F. Draw F B, which is another side of the required octagon. Bisect the two sides now obtained and erect perpendiculars to their middle points, G and H, which produce until they intersect at the point C. C then is the center of the circle that will circumscribe the octagon. From C as center, and with C B as radius, strike the circle, as shown. Set the dividers to the space A B and step off the circumference, obtaining the points L, K, M, O and N. Draw the connecting arcs A L, L K, K M, M O, O N and N F, thus completing the required figure.

Fig. 170.   Upon a Given Side to Draw a Regular Nonagon.

Fig. 170. - Upon a Given Side to Draw a Regular Nonagon.

Fig. 171   Upon a Given Side, to Draw a Regular Decagon.

Fig. 171 - Upon a Given Side, to Draw a Regular Decagon.

Fig. 172.   Upon a Given Side to Draw a Regular Undecagon.

Fig. 172. - Upon a Given Side to Draw a Regular Undecagon.

41. Upon a Given Side to Draw a Regular Nonagon. - In Fig. 170, A B is any given side upon which it is required to draw a regular nonagon. Produce A B indefinitely in the direction of D. From B as center, and with B A as radius, strike the semicircle A F D. At the point B erect a perpendicular to A B, cutting the semicircle in the point F. Draw the chord F D, which bisect obtaining the point G. From D as center, and with D G as radius, cut the semicircle in the point E. Draw E B, which will be another side of the required figure. From the middle points of the two sides now obtained, as H and K, erect perpendiculars, which produce until they intersect at the point C. Then C is the center of the circle which will circumscribe the required nonagon. From C as center, and with C B as radius, strike the circle B O P A. Set the dividers to the space A B and step off the circle, as shown, obtaining the points N, P, M, R, O and L. Draw the connecting chords, A N, N P, P M, M R, R O, O L and L E, thus completing the figure.

42. Upon a Given Side to Draw a Regular Decagon. - In Fig. 171, A B is the given side upon which a regular decagon is to be drawn. Produce A B indefinitely in the direction of D. From B as center, and with B A as radius, strike the semicircle A H D. Biscct the given side A B, obtaining the point F. Through the point B draw the line H B G, perpendicular to A B. From B as center, and with B F as radius, strike the arc F G, cutting the perpendicular H G in the point G. From G as center, and with G D as radius, strike the arc D O, cutting the perpendicular H G in the point O. From D as center, and with D O as radius, strike the arc O K, cutting the semicircle in the point K. Draw the line K D, which bisect with the line B L, cutting the semicircle in the point E. Then E B will be another side of the decagon. Upon the middle points, F and M, of the two sides now obtained erect perpendiculars, which produce until they intersect at the point C. Then C is the center of the circle which will circumscribe the required decagon. From C as center, and with C B as radius, strike the circle, as shown. Set the dividers to the space A B and step off the circle, obtaining the several points, I, N, S, V, R, T and P. Draw the connecting lines, A I, I N, N S. S V, V R, R T, T P and P E, thus completing the figure.