Problems in Geometrical Drawing.
[Illustration: 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_.
[Illustration: Fig. 100.]
[Illustration: 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_.
[Illustration: 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_.
[Illustration: Fig. 103.]
[Illustration: 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_.
[Illustration: Fig. 105.]
[Illustration: 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_.
[Illustration: 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.
[Illustration: Fig. 108.]
[Illustration: 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.
[Illustration: 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.
[Illustration: 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.
[Illustration: 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_.
[Illustration: Fig. 113.]
[Illustration: 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.
[Illustration: 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.
[Illustration: 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.
[Illustration: 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_.
[Illustration: Fig. 118.]
[Illustration: 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.
[Illustration: Fig. 120.]
[Illustration: Fig. 121.]