Problems in Geometrical Drawing.
Fig. 99.
Example 1.—To bisect (cut in two) a straight line or an arc of a circle, [Fig. 99]. From the ends of A B as centers, describe arcs cutting each other at C and D, and draw C D, which cuts the line at E or the arc at F.
Ex. 2.—To draw a perpendicular to a straight line, or a radial line to a circular arc, [Fig. 99]. Operate as in the foregoing problem. The line C D is perpendicular to A B; the line C D is also radial to the arc A B.
Fig. 100.
Fig. 101.
Ex. 3.—To draw a perpendicular to a straight line, from a given point in that line, [Fig. 100]. With any radius from any given point A in the line B C, cut the line at B and C. Next, with a longer radius, describe arcs from B and C, cutting each other at D, and draw the perpendicular D A.
Second Method, [Fig. 101]. From any center F above B C, describe a circle passing through the given point A, and cutting the given line at D; draw D F, and produce it to cut the circle at E; and draw the perpendicular A E.
Fig. 102.
Third Method, [Fig. 102]. From A describe an arc E C, and from E, with the same radius, the arc A C cutting the other at C; through C draw a line E C D and set off C D equal to C E, and through D draw the perpendicular A D.
Fig. 103.
Fig. 104.
Ex. 4.—To draw a perpendicular to a straight line from any point without it, [Fig. 103]. From the point A with a sufficient radius cut the given line at F and G; and from these points describe arcs cutting at E. Draw the perpendicular A E.
If there be no room below the line, the intersection may be taken above the line; that is to say, between the line and the given point.
Second Method, [Fig. 104]. From any two points B C at some distance apart, in the given line, and with the radii B A, C A, respectively, describe arcs cutting at A D. Draw the perpendicular A D.
Fig. 105.
Fig. 106.
Ex. 5.—To draw a parallel line through a given point, [Fig. 105]. With a radius equal to the given point C from the given line A B, describe the arc D from B, taken considerably distant from C. Draw the parallel through C to touch the arc D.
Second Method, [Fig. 106]. From A, the given point, describe the arc F D, cutting the given line at F; from F, with the same radius, describe the arc E A, and set off F D, equal to E A. Draw the parallel through the points A D.
Fig. 107.
When a series of parallels are required perpendicular to a base line A B, they may be drawn as in [fig. 107] through points in the base line set off at the required distances apart. This method is convenient also where a succession of parallels are required to a given line C D, for the perpendicular may be drawn to it, and any number of parallels may be drawn on the perpendicular.
Fig. 108.
Fig. 109.
Ex. 6.—To divide a line into a number of equal parts, [Fig. 108].
To divide the line A B into, say, five parts. From A and B draw parallels A C, B D on opposite sides; set off any convenient distance four times (one less than the given number), from A on A C, and on B on B D; join the first on A C to the fourth on B D, and so on. The lines so drawn divide A B as required.
Second Method, [Fig. 109]. Draw the line at A C, at an angle from A, set off, say, five equal parts; draw B 5, and draw parallels to it from the other points of division in A C. These parallels divide A B as required.
Fig. 110.
Ex. 7.—Upon a straight line to draw an angle equal to a given angle, [Fig. 110]. Let A be the given angle and F G the line. With any radius from the points A and F, describe arcs D E, I H, cutting the sides of the angle A and the line F G.
Set off the arc I H, equal to D E and draw F H. The angle F is equal to A as required.
Fig. 111.
Ex. 8.—To bisect an angle, [Fig. 111]. Let A C B be the angle; on the center C cut the sides at A B. On A and B as centers describe arcs cutting at D dividing the angle into two equal parts.
Fig. 112.
Ex. 9.—To find the center of a circle or of an arc of a circle. [Fig. 112]. Draw the chord A B, bisect it by the perpendicular C D, bounded both ways by the circle; and bisect C D for the center G.
Fig. 113.
Fig. 114.
Ex. 10.—Through two given points to describe an arc of a circle with a given radius, [Fig. 113]. On the points A and B as centers, with the given radius, describe arcs cutting at C; and from C, with the same radius, describe an arc A B as required.
Second, for a circle or an arc, [Fig. 114]. Select three points A, B, C in the circumference, well apart; with the same radius describe arcs from these three points cutting each other, and draw two lines D E, F G, through their intersections according to [Fig. 107]. The point where they cut is the center of the circle or arc.
Ex. 11.—To describe a circle passing through three given points, [Fig. 114]. Let A, B, C be the given points and proceed as in last problem to find the center O, from which the circle may be described.
This problem is variously useful; in finding the diameter of a large fly-wheel, or any other object of large diameter when only a part of the circumference is accessible; in striking out arches when the span and rise are given, etc.
Fig. 115.
Ex. 12.—To draw a tangent to a circle from a given point in the circumference, [Fig. 115]. From A set off equal segments A B, A D, join B D and draw A E, parallel to it, for the tangent.
Fig. 116.
Ex. 13.—To draw tangents to a circle from points without it, [Fig. 116]. From A with the radius A C describe an arc B C D, and from C with a radius equal to the diameter of the circle, cut the arc at B D, join B C, C D, cutting the circle at E F, and draw A E, A F, the tangents.
Fig. 117.
Ex. 14.—Between two inclined lines to draw a series of circles touching these lines and touching each other, [Fig. 117]. Bisect the inclination of the given lines A B, C D by the line N O. From a point P in this line draw the perpendicular P B to the line A B, and on P describe the circle B D, touching the lines and cutting the center lines at E. From E draw E F perpendicular to the center line, cutting A B at F, and from F describe an arc E G, cutting A B at G. Draw G H parallel to B P, giving H, the center of the next circle, to be described with the radius H E, and so on for the next circle, I N.
Fig. 118.
Fig. 119.
Ex. 15.—To construct a triangle on a given base, the sides being given.
First. An equilateral triangle, [Fig. 118]. On the ends of a given base A B, with A B as a radius describe arcs cutting at C, and draw A C, C B.
Second. Triangle of unequal sides, [Fig. 119]. On either end of the base A D, with the side B as a radius describe an arc; and with the side C as a radius, on the other end of the base as a center, describe arcs cutting the arc at E; join A E, D E.
This construction may be used for finding the position of a point C or E at given distances from the ends of a base, not necessarily to form a triangle.
Fig. 120.
Fig. 121.
Ex. 16.—To construct a square rectangle on a given straight line.
First. A square, [Fig. 120]. On the ends B A as centers, with the line A B as radius, describe arcs cutting at C; on C describe arcs cutting the others at D E; and on D and E cut these at F G. Draw A F, B G and join the intersections H I.
Second. A rectangle, [Fig. 121]. On the base E F draw the perpendiculars E H, F G, equal to the height of the rectangle, and join G H.
Fig. 122.
Ex. 17.—To construct a parallelogram of which the sides and one of the angles are given, [Fig. 122]. Draw the side D E equal to the given length A, and set off the other side D F equal to the other length B, forming the given angle C. From E with D F as radius, describe an arc, and from F, with the radius D E cut the arc at G. Draw F G, E G. Or, the remaining sides may be drawn as parallels to D E, D F.
Fig. 123.
Ex. 18.—To describe a circle about a triangle, [Fig. 123]. Bisect two sides A B, A C of the triangle at E F, and from these points draw perpendiculars cutting at K. On the center K, with the radius K A draw the circle A B C.
Fig. 124.
Ex. 19.—To describe a circle about a square, and to inscribe a square in a circle, [Fig. 124].
First. To describe the circle. Draw the diagonals A B, C D of the square, cutting at E; on the center E with the radius E A describe the circle.
Second. To inscribe the square. Draw the two diameters A B, C D at right angles and join the points A B, C D to form the square.
In the same way a circle may be described about a triangle.
Fig. 125.
Ex. 20.—To inscribe a circle on a square, and to describe a square about a circle, [Fig. 125].
First. To inscribe the circle. Draw the diagonals A B, C D of the square, cutting at E; draw the perpendicular E F to one side, and with the radius E F describe the circle.
Second. To describe the square. Draw two diameters A B, C D at right angles, and produce them; bisect the angle D E B at the center by the diameter F G, and through F and G draw perpendiculars A C, B D, and join the points A D and B C where they cut the diagonals to complete the square.
Fig. 126.
Ex. 21.—To inscribe a circle in a triangle, [Fig. 126]. Bisect two of the angles A C of the triangle by lines cutting at D; from D draw a perpendicular D E to any side, and with D E as radius describe a circle.
Fig. 127.
Ex. 22.—To inscribe a pentagon in a circle, [Fig. 127]. Draw two diameters A C, B D at right angles cutting at O; bisect A O at E, and from B with radius B E cut the circumference at G H and with the same radius step round the circle to I and K; join the points to form the pentagon.
Fig. 128.
Ex. 23.—To construct a hexagon upon a given straight line, [Fig. 128]. From A and B, the ends of the given line, describe arcs cutting at G; from G with the radius G A describe a circle. With the same radius set off the arcs A C, C F and B D, D E; join the points so found to form the hexagon.
Fig. 129.
Ex. 24.—To inscribe a hexagon in a circle, [Fig. 129]. Draw a diameter A C B; from A and B as centers, with the radius of the circle A C cut the circumference at D, E, F, G, and draw A D, D E, etc., to form the hexagon. The points D E, etc., may be found by stepping the radius (with the dividers) six times round the circle.
Fig. 130.
Ex. 25.—To describe an octagon on a given straight line, [Fig. 130]. Produce the given line A B both ways and draw perpendiculars A E, B F; bisect the external angles A and B by the lines A H, B C, which make equal to A B. Draw C D and H G parallel to A E and equal to A B; from the center G D, with the radius A B, cut the perpendiculars at E F, and draw E F to complete the hexagon.
Fig. 131.
Ex. 26.—To convert a square into an octagon, [Fig. 131].—Draw the diagonals of the square cutting at E; from the corners A, B, C, D, with A E as radius, describe arcs cutting the sides at G, H, etc., and join the points so found to complete the octagon.
Fig. 132.
Ex. 27.—To inscribe an octagon in a circle, [Fig. 132]. Draw two diameters A C, B D, at right angles; bisect the arcs A B, B C, at E, F, etc., to form the octagon.
Fig. 133.
Ex. 28.—To describe an octagon about a circle, [Fig. 133]. Describe a square about the given circle A B, draw perpendiculars H and K, to the diagonals, touching the circle to form the octagon. Or, the points H, K, etc., may be found by cutting the sides from the corners, by lines parallel to the diagonals.
Fig. 134.
Ex. 29.—To describe an ellipse when the length and breadth are given, [Fig. 134]. On the center C, with A E as radius, cut the axis A B at F and G, the foci, fix a couple of pins into the axis at F and G, and loop on a thread or cord upon them equal in length to the axis A B, so as when stretched to reach the extremity C of the conjugate axis, as shown in dot-lining. Place a pencil or drawpoint inside the cord, as at H, and guiding the pencil in this way, keeping the cord equally in tension, carry the pencil round the pins F, G, and so describe the ellipse.
Note.—The ellipse is an oval figure, like a circle in perspective. The line that divides it equally in the direction of its great length is the transverse axis, and the line which divides the opposite way is the conjugate axis.
Second Method. Along the straight edge of a piece of stiff paper mark off a distance a c equal to A C, half the transverse axis; and from the same point a distance a b equal to C D, half the conjugate axis. Place the slip so as to bring the point b on the line A B of the transverse axis, and the point c on the line D E; and set off on the drawing the position of the point a. Shifting the slip, so that the point travels on the transverse axis, and the point c on the conjugate axis, any number of points in the curve may be found, through which the curve may be traced. See [fig. 135].
Fig. 135.