Problems in Geometrical Drawing.
LinesFig. 99.
Fig. 99.
Example1.—To bisect(cut in two)a straight line or an arc of a circle,Fig. 99. From the ends ofA Bas centers, describe arcs cutting each other atCandD, and drawC D, which cuts the line atEor the arc atF.
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 lineC Dis perpendicular toA B; the lineC Dis also radial to the arcA B.
PerpendicularsFig. 100.
Fig. 100.
ExampleFig. 101.
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 pointAin the lineB C, cut the line atBandC. Next, with a longer radius, describe arcs fromBandC, cutting each other atD, and draw the perpendicularD A.
Second Method,Fig. 101. From any centerFaboveB C, describe a circle passing through the given pointA, and cutting the given line atD; drawD F, and produce it to cut the circle atE; and draw the perpendicularA E.
ExampleFig. 102.
Fig. 102.
Third Method,Fig. 102. FromAdescribe an arcE C, and fromE, with the same radius, the arcA Ccutting the other atC; throughCdraw a lineE C Dand set offC Dequal toC E, and throughDdraw the perpendicularA D.
ExampleFig. 103.
Fig. 103.
ExampleFig. 104.
Fig. 104.
Ex.4.—To draw a perpendicular to a straight line from any point without it,Fig. 103. From the pointAwith a sufficient radius cut the given line atFandG; and from these points describe arcs cutting atE. Draw the perpendicularA 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 pointsB Cat some distance apart, in the given line, and with the radiiB A,C A, respectively, describe arcs cutting atA D. Draw the perpendicularA D.
ExampleFig. 105.
Fig. 105.
ExampleFig. 106.
Fig. 106.
Ex.5.—To draw a parallel line through a given point,Fig. 105. With a radius equal to the given pointCfrom the given lineA B, describe the arcDfromB, taken considerably distant fromC. Draw the parallel throughCto touch the arcD.
Second Method,Fig. 106. FromA, the given point, describe the arcF D, cutting the given line atF; fromF, with the same radius, describe the arcE A, and set offF D, equal toE A. Draw the parallel through the pointsA D.
ExampleFig. 107.
Fig. 107.
When a series of parallels are required perpendicular to a base lineA B, they may be drawn as infig. 107through 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 lineC D, for the perpendicular may be drawn to it, and any number of parallels may be drawn on the perpendicular.
ExampleFig. 108.
Fig. 108.
ExampleFig. 109.
Fig. 109.
Ex.6.—To divide a line into a number of equal parts,Fig. 108.
To divide the lineA Binto, say, five parts. FromAandBdraw parallelsA C,B Don opposite sides; set off any convenient distance four times (one less than the given number), fromAonA C, and onBonB D; join the first onA Cto the fourth onB D, and so on. The lines so drawn divideA Bas required.
Second Method,Fig. 109. Draw the line atA C, at an angle fromA, set off, say, five equal parts; drawB5, and draw parallels to it from the other points of division inA C. These parallels divideA Bas required.
ExampleFig. 110.
Fig. 110.
Ex.7.—Upon a straight line to draw an angle equal to a given angle,Fig. 110. LetAbe the given angle andF Gthe line. With any radius from the pointsAandF, describe arcsD E,I H, cutting the sides of the angleAand the lineF G.
Set off the arcI H, equal toD Eand drawF H. The angleFis equal toAas required.
ExampleFig. 111.
Fig. 111.
Ex.8.—To bisect an angle,Fig. 111. LetA C Bbe the angle; on the centerCcut the sides atA B. OnAandBas centers describe arcs cutting atDdividing the angle into two equal parts.
ExampleFig. 112.
Fig. 112.
Ex.9.—To find the center of a circle or of an arc of a circle.Fig. 112. Draw the chordA B, bisect it by the perpendicularC D, bounded both ways by the circle; and bisectC Dfor the centerG.
ExampleFig. 113.
Fig. 113.
ExampleFig. 114.
Fig. 114.
Ex.10.—Through two given points to describe an arc of a circle with a given radius,Fig. 113. On the pointsAandBas centers, with the given radius, describe arcs cutting atC; and fromC, with the same radius, describe an arcA Bas required.
Second, for a circle or an arc,Fig. 114. Select three pointsA,B,Cin the circumference, well apart; with the same radius describe arcs from these three points cutting each other, and draw two linesD E,F G, through their intersections according toFig. 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. LetA,B,Cbe the given points and proceed as in last problem to find the centerO, 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.
ExampleFig. 115.
Fig. 115.
Ex.12.—To draw a tangent to a circle from a given point in the circumference,Fig. 115. FromAset off equal segmentsA B,A D, joinB Dand drawA E, parallel to it, for the tangent.
Fig. 116.
Fig. 116.
Ex.13.—To draw tangents to a circle from points without it,Fig. 116. FromAwith the radiusA Cdescribe an arcB C D, and fromCwith a radius equal to the diameter of the circle, cut the arc atB D, joinB C,C D, cutting the circle atE F, and drawA E,A F, the tangents.
ExampleFig. 117.
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 linesA B,C Dby the lineN O. From a pointPin this line draw the perpendicularP Bto the lineA B, and onPdescribe the circleB D, touching the lines and cutting the center lines atE. FromEdrawE Fperpendicular to the center line, cuttingA BatF, and fromFdescribe an arcE G, cuttingA BatG. DrawG Hparallel toB P, givingH, the center of the next circle, to be described with the radiusH E, and so on for the next circle,I N.
ExampleFig. 118.
Fig. 118.
ExampleFig. 119.
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 baseA B, withA Bas a radius describe arcs cutting atC, and drawA C,C B.
Second. Triangle of unequal sides,Fig. 119. On either end of the baseA D, with the sideBas a radius describe an arc; and with the sideCas a radius, on the other end of the base as a center, describe arcs cutting the arc atE; joinA E,D E.
This construction may be used for finding the position of a pointCorEat given distances from the ends of a base, not necessarily to form a triangle.
ExampleFig. 120.
Fig. 120.
ExampleFig. 121.
Fig. 121.
Ex.16.—To construct a square rectangle on a given straight line.
First. A square,Fig. 120. On the endsB Aas centers, with the lineA Bas radius, describe arcs cutting atC; onCdescribe arcs cutting the others atD E; and onDandEcut these atF G. DrawA F,B Gand join the intersectionsH I.
Second. A rectangle,Fig. 121. On the baseE Fdraw the perpendicularsE H,F G, equal to the height of the rectangle, and joinG H.
ExampleFig. 122.
Fig. 122.
Ex.17.—To construct a parallelogram of which the sides and one of the angles are given,Fig. 122. Draw the sideD Eequal to the given lengthA, and set off the other sideD Fequal to the other lengthB, forming the given angleC. FromEwithD Fas radius, describe an arc, and fromF, with the radiusD Ecut the arc atG. DrawF G,E G. Or, the remaining sides may be drawn as parallels toD E,D F.
ExampleFig. 123.
Fig. 123.
Ex.18.—To describe a circle about a triangle,Fig. 123. Bisect two sidesA B,A Cof the triangle atE F, and from these points draw perpendiculars cutting atK. On the centerK, with the radiusK Adraw the circleA B C.
ExampleFig. 124.
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 diagonalsA B,C Dof the square, cutting atE; on the centerEwith the radiusE Adescribe the circle.
Second. To inscribe the square. Draw the two diametersA B,C Dat right angles and join the pointsA B,C Dto form the square.
In the same way a circle may be described about a triangle.
ExampleFig. 125.
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 diagonalsA B,C Dof the square, cutting atE; draw the perpendicularE Fto one side, and with the radiusE Fdescribe the circle.
Second. To describe the square. Draw two diametersA B,C Dat right angles, and produce them; bisect the angleD E Bat the center by the diameterF G, and throughFandGdraw perpendicularsA C,B D, and join the pointsA DandB Cwhere they cut the diagonals to complete the square.
ExampleFig. 126.
Fig. 126.
Ex.21.—To inscribe a circle in a triangle,Fig. 126. Bisect two of the anglesA Cof the triangle by lines cutting atD; fromDdraw a perpendicularD Eto any side, and withD Eas radius describe a circle.
ExampleFig. 127.
Fig. 127.
Ex.22.—To inscribe a pentagon in a circle,Fig. 127. Draw two diametersA C,B Dat right angles cutting atO; bisectA OatE, and fromBwith radiusB Ecut the circumference atG Hand with the same radius step round the circle toIandK; join the points to form the pentagon.
ExampleFig. 128.
Fig. 128.
Ex.23.—To construct a hexagon upon a given straight line,Fig. 128. FromAandB, the ends of the given line, describe arcs cutting atG; fromGwith the radiusG Adescribe a circle. With the same radius set off the arcsA C,C FandB D,D E; join the points so found to form the hexagon.
ExampleFig. 129.
Fig. 129.
Ex.24.—To inscribe a hexagon in a circle,Fig. 129. Draw a diameterA C B; fromAandBas centers, with the radius of the circleA Ccut the circumference atD,E,F,G, and drawA D,D E, etc., to form the hexagon. The pointsD E, etc., may be found by stepping the radius (with the dividers) six times round the circle.
ExampleFig. 130.
Fig. 130.
Ex.25.—To describe an octagon on a given straight line,Fig. 130. Produce the given lineA Bboth ways and draw perpendicularsA E,B F; bisect the external anglesAandBby the linesA H,B C, which make equal toA B. DrawC DandH Gparallel toA Eand equal toA B; from the centerG D, with the radiusA B, cut the perpendiculars atE F, and drawE Fto complete the hexagon.
ExampleFig. 131.
Fig. 131.
Ex.26.—To convert a square into an octagon,Fig. 131.—Draw the diagonals of the square cutting atE; from the cornersA,B,C,D, withA Eas radius, describe arcs cutting the sides atG,H, etc., and join the points so found to complete the octagon.
ExampleFig. 132.
Fig. 132.
Ex.27.—To inscribe an octagon in a circle,Fig. 132. Draw two diametersA C,B D, at right angles; bisect the arcsA B,B C, atE,F, etc., to form the octagon.
ExampleFig. 133.
Fig. 133.
Ex.28.—To describe an octagon about a circle,Fig. 133. Describe a square about the given circleA B, draw perpendicularsHandK, to the diagonals, touching the circle to form the octagon. Or, the pointsH,K, etc., may be found by cutting the sides from the corners, by lines parallel to the diagonals.
ExampleFig. 134.
Fig. 134.
Ex.29.—To describe an ellipse when the length and breadth are given,Fig. 134. On the centerC, withA Eas radius, cut the axisA BatFandG, the foci, fix a couple of pins into the axis atFandG, and loop on a thread or cord upon them equal in length to the axisA B, so as when stretched to reach the extremityCof the conjugate axis, as shown in dot-lining. Place a pencil or drawpoint inside the cord, as atH, and guiding the pencil in this way, keeping the cord equally in tension, carry the pencil round the pinsF,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 thetransverse axis, and the line which divides the opposite way is theconjugate axis.
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 thetransverse axis, and the line which divides the opposite way is theconjugate axis.
Second Method.Along the straight edge of a piece of stiff paper mark off a distancea cequal toA C, half the transverse axis; and from the same point a distancea bequal toC D, half the conjugate axis. Place the slip so as to bring the pointbon the lineA Bof the transverse axis, and the pointcon the lineD E; and set off on the drawing the position of the pointa. Shifting the slip, so that the point travels on the transverse axis, and the pointcon the conjugate axis, any number of points in the curve may be found, through which the curve may be traced. Seefig. 135.
ExampleFig. 135.
Fig. 135.