314
FUNDAMENTALS OF COLLEGE GEOMETRY5. The points QI> Q2, Q:J, . . . , Q,,-t divide AB into n congruent seg- ments.
A Proof
ST ATEMENTS REASONS
1. APt
=
PtP2=
P2P3= . . . = Pn-IP",
2. PIQt II P2~ II P3QJ II... II PI/B.3. AQI
= Qt~ = Q.2~='" = Q"-IB.
1. By construction.
2. By construction.
3. Theorem 6.8.
Construction 8
10.12. Circumscribe a circle about a triangle.
Given: LoABC.
To construct: Circumscribe a 0 about LoABC.
Construction:
Comtruction 8.
SL\TDIE,\;TS RF "SONS
B_-....
" ,
\ \
\ I
I I
A'
,
.~I / 'C" II /
---"""
1. Construction 4.
1. Construct the perpendicular bi- sectors of two sides of the Lo.
2. The two lines meet at a point O.
3. With () as the center and mOA as the radius construct the circle O.
4. 0 () is the circumscribed circle.
2. Theorem 3.1.
3. Postulate 19.
(The proof of this construction is left as an exercise.) Construction 9
10.13. To inscribe a circle in a given triangle.
Given: LoABC.
To construct: Inscribe a circle in LoABC.
Construction 9.
B
Construction:
STATEMENTS REASONS
1. Bisect two angles of LoABC.
2. Let 0 be the point of intersection of the bisectors.
3. Construct OD from 0 perpen- dicular to AC.
4. With 0 as center and mOD as as radius, construct 0 O.
:'J. 00 is the inscribed O.
1. Construction 2.
2. Theorem 3.1.
3. Construction 5.
4. Postulate 19.
(The proof of this construction is left as an exercise.) Exercises (B)
1. Draw two points 3 inches apart. Locate by construction all the points 2 inches horn each of these given points.
2. Draw two lines intersecting at 45° and two other parallel lines which are ] inch apart. Locate by construction all the points equidistant from the intersecting lines and equidistant from the parallel lines.
:3. Draw two lines /1 and /2 intersecting at a 60° angle. Locate by construction all the points that are 1 inch from II and 12,
4. Draw a circle 0 with a radius length of 2 inches. Draw a diameter AB.
Lucale by cunstruction the puints that are I inch from the diameter AB and equidistant horn A and O.
:'J. Draw a triangle ABC with measures of the sides equal to 2 inches, 2t inches, and 3 inches. Locate by construction the points on the altitude from B that are equi- distant from Band C.
6. In the triangle of Ex. 5, locate by construction all the points on the altitude from C that are equi- distant from sides AB and BC.
7. In the triangle of Ex. 5, locate by construction all the points on the median from C that are equidistant from A and C.
~. In the triangle of Ex. 5, locate by construction all the points equidistant from sides AC and BC at a distance oft inch from side AB.
9. Draw a triangle ABC. Locate by construction the point P that is equi- distant from the vertices of the triangle. With P as center and mPB as
c
Q
A 3 B
Ex.\'. 5-8.
316
radius, construct a circle. (This circle is said to be circumscribed about the triangle.)
10. Draw a triangle ABC. Locate by construction the point P equidistant from the sides
2I
the triangle. Construct the segment PM from P per- pendicular to AB. With P as center and mPM as radius, construct a circle. (This circle is said to be inscribed in the triangle.)Summary Test
Constructions Test
1-4. With ruler and protractor draw AB
=
3 inches and La whose measure is 40.]. Construct an isosceles triangle with base equaling AB and base angle with measure equaling mLa.
2. Construct an isosceles triangle with leg equaling AB and vertex angle with measure equaling mLa.
j. Construct an isosceles triangle with altitude to the base equaling AB and vertex angle with measure equaling mLa.
4. Construct an isosceles triangle with base equaling AB and vertex angle with measure equaling mLa.
5-9. With ruler and protractor draw segments AB = 2 inches, CD = 3 inches, EF = 4 inches, and La whose measure is 40.
5. Construct DPQRS with PS = AB, PQ
= CD and PR = EF.
6. Construct DPQRS with PS = AB, PR = EF, and SQ = CD.
7. Construct DPQRS with PS = AB, PR = CD, and LSPR
== La.
8. ConstructD PQRS with PQ
= AB, PR = EF, and altitude on PQ = CD.
~J. ConstructD PQRS with PQ
= CD, LSPQ
== La, and altitude on PQ = AB.
]0. Construct an angle whose measure is 75.
]I. Draw a line 5 inches long. Then divide it into five congruent segments using only a compass and straight edge.
317
Then construct the three medians of the triangle.
struct perpendiculars to the three sides of f:..ABC.
13. Draw an obtuse triangle. Then construct a circle which circumscribes the '~
triangle.:i
14. Dra~ a triangle. Then construct a circle which is inscribed in the triangle.~1 15. Draw an obtuse triangle. Then construct the three altitudes of the
triangle.
16. Draw a triangle.
318
.
.11
II
~.
Ill/
Geometric Loci
11.1. Loci and sets. The set of all points is space. A geometric figure is a set of points governed by one or more limiting geometric conditions. Thus, a geometric figure is a subset of space.
In Chapter 7 we defined a circle as a set of points lying in a plane which are equidistant from a fixed point of the plane.
Mathematicians sometimes use the term "locus" to describe a geometric figure.
-
Definition: A locus of points is the set of all the points, and only those points, which satisfy one or more given conditions.
Thus, instead of using the words "the set of points P such that. . . ," we could say "the locus of points P such that. . . ." A circle can be defined as the
"locus of points lying in a plane at a given distance from a fixed point of the plane."
Sometimes one will find the locus defined as the path of a point moving according to some given condition or set of conditions.
Consider the path of the hub of a wheel that moves along a level road (Fig. 11.1). A, B, C, D represent positions of the center of the wheel at different instants during the motion of the wheel. It should be evident to the reader that, as the wheel rolls along the road, the set of points which re- present the positions of the center of the hub are elements of a line parallel to the road and at a distance from the road equal to the radius of the wheel.
We speak of this line as "the locus of the center of the hub of the wheel as the wheel moves along the track." In this text locus lines will be drawn with
319
320
FUNDAMENTALS OF COLLEGE GEOMETRY-8f?faA-
Fig. 11.1.
from given and construction
--
QIQ2/
~
/
~4
I \
Pit .0 ~Q
\ I
4
\ h3
~ /
'--.
P2-
Q3/Fig. 11.2.
long dash lines to distinguish them As a second simple illustration of a locus, consider the problem of finding the locus of points in a plane 2 inches from a given point 0 (Fig. 11.2).
Let us first locate several points, such as Pt, P2, P3, P4,..., which are 2 inches from O. Obviously there are an infinite number of such points.
Next draw a smooth curve through these points. In this case it appears that the locus is a circle with the cen- ter at () and a radius whose measure is 2 inches.
If now, conversely, we select points such as Qt, Q2' Q;j, Q4' . . . , each om"
of which meets the requirement of being 2 inches from 0, it is evident thai' me circle.
Thus, to prove that a line is a locus, it is necessary to prove the followmsll. i two characteristics:'
I
'I. A ny point on the line
satisfies the given condition or set of conditions. '.' '2. fj thu (a) any paint that wNlji" th, ginn. wnditian
"' ,,' of ",nditia'"
" ""~.
,:
the line or (b) any point not on the line does not satisfy the condition. '~
The word locus (plural loci, pronounced "10' -si") is the Latin word mea~in(
"place" or "location." A locus may consist of one or more points, hnes,;
surfaces, or combinations of these. '
11.2. Determining a locus. Let us use the example of Fig.
the general method of determining a locus.
Step I: Locate several points which satisfy the given conditions.
Step II: Draw a smooth line or lines (straight or curved) through these points. MI
"., '
, '
,
':
Step III: Form a conclusion as to the locus, and describe accurately the geome.]