LMBE
12. Find mAB if mER = 5 and mRF = 7
Test 3
204 FUNDAMENTALS OF COLLEGE GEOMETRY
27. A trapezoid is equilateral if it has two congruent sides.
28. The sum of the measures of the angles of a quadrilateral is 360.
29. The bisectors of the opposite angles of a rectangle are parallel.
30. The bisectors of the adjacent angles of a parallelogram are perpendicular.
PROBLEMS
I. AC is the diagonal of rhombus ABCD. If mLB
= 120, find mLBAC.
2. InD ABCD, mAB = 10 inches, mLB = 30, and AH ..1 BC. Find mAH.
3. InD ABCD, mLA = 2mLB. Find mLA.
4. In 0 ABCD, diagonal AC ..1 BC and AC ==
BC. Find mLD.
5. In LABC, AD
==DB; mLC = 90; mLB = 30; mAC = 14 inches. Fin mBD.
6. In LABC, AD
==DB; mLC = 90; mLA = 60; mCD = 12 inches. Fin' mAC.
7. In LABC, AD
==DB; mLC = 90; mLA = 60; mAB = 26 inches.
mCD.
A
I
i .
c
2.Given: AM is a median of LABC; LDCB
~----/71
Circles
7.1. Uses of the circle.
The history of civilization's continually improving
Iconditions of living and working is intimately related to the use of the pro- I
.~
perties of the circle. One of the most important
applications of the circle) that man has invented is the wheel. Without the wheel most of the world's' work would cease. Industry would be completely crippled withoUt the circle in the form of wheels, gears, and axles.
Transportation would revert back to conditions of prehistoric times. Without the wheel there would be no bicycles, no automobiles, no trains, no streetcars, and no airplanes. Farm machines, factory and mine equipment, without the wheel, would exist 0 in the form of useless metal, plastics, and wood.
Industry applies the properties of the circle when it uses ball bearings to, reduce friction and builds spherical tanks for
strength. Every year miJIions1 of feet of circular pipe and wire are manufactured.
Countless manufac- tured articles of furniture, dishes, and tools are circular in form. Most ofthe cans of food on the grocery shelves have circular cross sections. Tanks I with circular cross sections have many uses (see Fig. 7.1).
Circular shapes are found in such ornamental designs as rose windows, architectural columns, traffic circles, and landscape designs.
7.2. The circle in history. The invention and use of the circular wheel date
back to very early times. No one knows when the wheel was invented or
~
who invented it. Some authorities believe that the wheel was invented some- j where in Asia about 10,000 years ago. The oldest wheel in existence was discovered in Mesopotamia in 1927 when archeologists uncovered a four- wheeled chariot known to have existed about 5,500 years ago (Fig. 7.2).
The circle had an aesthetic appeal to the Greeks. To them it was the most,:.
r~
L
Fig. 7.1. The world's largest sewage treatment system at Chicago has been termed by the American Society oj Civil Engineers "the seventh wondRr of
America." More than 1,100,000,000 gallons of waste are treated rinily ill the system. In this view can be seen the preliminary settling tanks, the aeration tanks, and thefmal settlillg tanh. Each of the final settling tanks is 126jeet in diameter. (Chicago Aerial Industries, Inc.)
~"~
~'c';~.,::,-,":~
~:'g. 72. View oj one side of a complele chariot found in a brick tomb 20feel below plain level at Kish.
"}rca 4000 B.c. Skeleton of one of the oxen appears in original position beside the pole. (Chicago Aatural History iWuseum.)
207
208
FUNDAMENTALS OF COLLEGE GEOMETRYperfect of all plane figures. Thales, Pythagoras, Euclid, and Archimedes each contributed a great deal to the geometry of the circle.
!
Thales probably is best known for the deductive character of his geometric propositions. One of the most remarkable of his geometrical achievement~
was proving that any angle inscribed in a semicircle must be a right anglt'j
(§ 7.16). I
Pythagoras was the founder of the Pythagorean school, a brotherhood 0 '
people with common philosophical and political beliefs. They were boun by oath not to reveal the teachings or secrets of the school. Pythagoras w
'primarily a philosopher. Members of his school boasted that they sough knowledge, not wealth. They were probably the first to arrange the variou propositions on geometric figures in a logical order. No attempt at fir:' was made to apply this knowledge to practical mechanics. Much of th~
early work on geometric constructions involving circles was attributed to thij
group. .~
:
Euclid published systematic, rigorous proofs of the leading propositio~
of the geometry known at his time. His treatise, entitled "Elements," was.
to a large extent, a compilation of works of previous philosophers an.
mathematicians. However, the form in which the propositions was pre~
sented, consisting of statement, construction, proof and conclusion, was th~
work of Euclid. Much of Euclid's work was done when he served as a teacheJ1j in Alexandria. It is probable that his "Elements" was written to be used as a"
text in schools of that time. The Greeks at once adopted the work as their~
standard textbook in their studies on pure mathematics. Throughout the;
-~. ~- 4. .. . .
i
most other textbooks in this field"
I
Archimedes, like his contemporaries, held that it was undesirable for a' '
philosopher to apply the results of mathematical science to any practical use.
.
'
. .
However, he did introduce a number of new inventions. ,,
Most readers are familiar with the story of his detection of the fraudulent!, goldsmith who diluted the gold in the king's crown. The Archimedean:
screw was used to take water out of the hold of a ship or to drain lands' inundated by the floodwaters of the Nile. Burning glasses and mirrors to, destroy enemy ships and large catapults to keep Romans besieging Syracuse;
at bay are devices attributed to the remarkable mechanical ingenuity of this;
m~. j
Science students today are referred to Archimede's principles dealing with!
the mechanics of solids and fluids. His work in relating the radius and the,1 circumference of a circle and in finding the area of the circle has stood the,!
test of time.
It is told that Archimedes was killed by an enemy soldier while studyin~j geometric designs he had drawn in the sand.
7.3. Basic definitions. To develop proofs for various theorems on circles, we must have a foundation of definitions and postulates. Many of the terms to be defined the student will recognize from his previous studies in mathe- matics.
A circle is the set of points lying in one plane each of which is equidistant from a given point of the plane. The given point is called the center of the circle (Fig. 7.3). Circles are often drawn with a compass (Fig. 7.4). The symbol for circle is O. In Fig. 7.3,0 is the center of OABC, or simply 00.
A line segment one of whose end- points is the center of the circle and the other one a point on the circle is a radius (plural, radii) of the circle. OA, DB, and OC are radii of 00. Thus, we can say radii of the same circle are congruent.
A chord of a circle is a segment whose endpoints are points of the circle. A
iameler IS a chord contaInIng- the center of the circle. ED is a chord of the circle in Fig. 7.3.
It will be noted that we defined
"radius" and "diameter" as a segment;
that is, as a set of points. Common usage, however, often lets the words de- note their measures. Thus we speak of a circle with radius of, say, 7 inches. Or We speak of a diameter equaling twice the radius. No confusion should arise because the context of the statement should clearly indicate whether a set of points or a number is being referred to.
Circles are congruent iff they have Congruent radii. Concentric circles are coplanar circles having the same center and noncongruent radii (see Fig. 7.5).
A
Fig. 7.4.
CIRCLES 209 D
0 B
Fig. 7.3.
1\\1
,
210
A
Fig. 7.5. Concurrent circles. Fig. 7.6. Inscribed polygon.
A circle is said to be circumscribed about a polygon when it contains all th vertices of the polygon. In Fig. 7.6, the circle is circumscribed about polygoJ ABCDE.
A polygon is said to be inscribed in a circle if each of its vertices lies on th circle. Thus its sides will be chords of the circle. Polygon ABCDE is in~~
scribed in the circle. J-
The interior of a circle is the union of its center and the set of all points
I
the plane of the circle whose distances from the center are less than the radiu$J
The "",.;'" of a ,;,de ;, the ,,' of po;,,,, ;n the plane of the dcde ,uch thaj~
their distances from the center are greater than the radius. Frequentl
l
"the words "inside" and "outside" are used for "interior" and "exterior.". I
81
!--
7.4. Tangent. Secant. A line is iangentto "a Circlertiiliesln"lhe- --- plane of the circle and intersects it in only one point. This point is called the point of tangency, and we say that the line and the circle are tangent at this point. In Fig. 7.7,