The Construction Of Regular Polygons II By The Use 188

Fig. 185.

Fig. 186. To Inscribe a Square within a Given Circle.

Fig. 186. To Inscribe a Square within a Given Circle.

The Construction Of Regular Polygons II By The Use 190

Fig. 187.

Fig. 188. To Inscribe a Regular Hexagon within a Given Circle.

Fig. 188. To Inscribe a Regular Hexagon within a Given Circle.

5l. To Inscribe an Octagon within a Given Circle. In Fig. 189, let K be the center of the given circle. Place a 45-degree set-square as shown in the engraving, bringing its long side in contact with the center, and mark the points E and A. Keeping it in the same position, move it along until its vertical side is in contact with K and mark the points D and H. Reverse the set-square from the position shown in the engraving, and mark the points C and G. Move the T-square upward until it touches the point K, and mark the points

B and F. Then A, H, G, F, E, D, C and B are corners of the octagon. The figure may now be readily completed by drawing the sides, by means of these points, using any rule or straight-edge for the purpose, all as shown by A H, H G, G F, F E, E D, D C, C B and B A, or by means of a 22 1/2 X 67 1/2-degree set-square.

Fig. 189.   To Inscribe a Regular Octagon within a Given Circle.

Fig. 189. - To Inscribe a Regular Octagon within a Given Circle.

52. To Draw an Equilateral Triangle upon a Given

Side. - In Fig. 190, let A B be the given side. First bring the line A B at right angles to the blade of the T-square. Then set the edge C B of a 30-degree set-square against the T-square, and move it along until the edge B D meets the point B, and draw the line B F. Reverse the set-square, still keeping the side C B against the T-square, and move it along until the side B D meets the point A, and draw the line A F, thus completing the figure.

Fig. 190.   To Draw an Equilateral Triangle upon a Given Side.

Fig. 190. - To Draw an Equilateral Triangle upon a Given Side.

53. To Draw a Square upon a Given Side. - In Fig. 191, let A B be the given side drawn at right angles to the blade of the T-square. Set the edge E F of a 45-degree set-square against the T-square, as shown, and move it along until the side E G meets the point B, and draw B I indefinitely. Reverse the set-square, and, bringing the side E G against the point A, draw A

Fig. 191.   To Draw a Square upon a Given Side.

Fig. 191. - To Draw a Square upon a Given Side.

F indefinitely. Bring the T-square against the point B and draw B F, producing it until it meets the line A F in the point F. In like manner draw A I, meeting the line B I in the point I. Then, with the set-square placed as shown in the engraving, connect I and F, thus completing the required figure.

54. To Draw a Regular Hexagon upon a Given Side. - In Fig. 192, let A B be the given side in a vertical position. Set the edge G H of a 30-degree set-square against the T-square, as shown, and move it along until the edge I G coincides with the point A, and draw the line A D indefinitely. Reverse the set-square, still keeping the edge G H against the T-square, and move it along until the side I G coincides with the point B, and draw B E indefinitely. These lines will intersect in the point 0, which will be the center of the required figure. Still keeping the edge G H of the set-square against the T-square, move it along until the perpendicular edge I H meets the point O, and through O draw F C indefinitely. With the set-square in the position shown in the engraving slide it along until the edge I G meets the point B, and draw B C, producing it until it meets the line F C in the point C. Reverse the set-square, still keeping the edge G H against the T-square, and draw the line C D, producing it until it meets the line A D in the point D. Slide the set-square along until the side I H meets the point D, and draw the line D E, meeting the line B E in the point E. Move the set-square along until the edge I G meets the point A, and draw the line A F, meeting the line C F in the point F. Now bring the set-square to its first position and slide it along until the edge I G meets the points F and E, and draw F E, thus completing the required figure. 55. To Draw a Regular Octagon upon a Given Side-In Fig. 193, let C D be the given side, drawn perpendicular to the blade of the T-square. Place one of the short sides of a 45-degree set-square against the T-square, as shown in the engraving. Move the set-square along until its long side coincides with the point C. Draw the line C B, and make it in length equal to C D. With the T-square draw the line A B, also in length equal to C D. Reverse the set-square, and bring the edge against the point A. Draw A II in length the same as C D. Still keeping a short side of the set-square against the T-square, slide it along until the other short side meets the point H, and draw H G, also of the same length. Then, using the long side of the set-square, draw G F of corresponding length. By means of the T-square draw F E, and by reversing the set-square draw E D, both in length equal to the original side, C D, joining it in the point I), thus completing the required octagon.