LBPO
I. Secants ABP and CDP intersect at P outside 0 O
2. Draw AD.
3. mLP + mLf3 = mLa.
4. mLP = mLa - mLf3.
5. mLa
=
fmAC; mLf3= tmBD.
6. mLa-mLf3 = HmAC-mBD].
7. mLP
= HmAC-mBD].
--
1. Given.
2. Postulate 2.
3. Theorem 5.16; symmetric pro- perty.
4. Subtraction property.
5. Theorem 7.3.
6. Subtraction property.
7. Substitution property.
238
7.35. Corollary: The measure of the angle formed by a secant and a tangent intersecting outside a circle is half the difference of the measures of the intercepted arcs.
7.36. Corollary: The measure of the angle formed by two tangents drawn from an external point to a circle is half the difference of the measures of the intercepted arcs.
Exercises
c
A Corollary 7.36.
Find the number of degrees measure in La, Lf3, and in arc s. 0 is the center of, a circle.'
Ex. I.
Ex.3.
D
~
c i
Ex. 2.
D D
c
Ex. 4.
A
a:>
Ex. 5.
Ex. 7.
p
Ex. 9.
0 0 N r
A
B
~B
A
D Ex. 6.
80°
A B
Ex. 8. CDIIAB.
s
R p
Ex. 10.
240
FUNDAMENTALS OF COLLEGE GEOMETRYD
0lC)
,,j
A p
Ex. 11. Ex.12.
Ex.l3. Ex. 14.
c
A
iF' c
Ex.15. Ex.16.
l
,
. .
i,'\',
I
',-
-
I
p
c
p 0
Lf')
Ex.17.
a
'"
.-...
Ex.19.
CIRCLES 241 A
p
Ex. IS.
A
Ex. 20.
Summary Tests
Test 1
COMPLETION STATEMENTS
1. In 00 the diameterAOB and tangent AT are-.
2. A central angle of a circle is formed by two
-'3. An inscribed angle of a circle is formed by two -' 4. An angle inscribed ina semicircle is a(n) ~ angle.
5. The greatest number of obtuse angles an inscribed triangle can IS_.
6. Tangent segments drawn to a circle from an outside point are_.
7. The largest chord of a circle is the
-
of the circle.8. An angle is inscribed in an arc. If the intercepted arc is increased by the inscribed angle is increased by-'
9. The opposite angles of an inscribed quadrilateral are-.
10. A line through the center of a circle and perpendicular to a chord - the chord and its arc.
11. If a line is - to a radius at its point on the circle, it is tangent to circle.
12. If two circles intersect, the line joining their centers is the
-
of tcommon chord.
13. In a circle, or in congruent circles, chords equidistant from the center circle are_.
14. An angle formed by two tangents drawn from an external point 1 circle is equal in degrees to one-half the - of its intercepted arcS.
242
TRUE-FALSESTATEMENTS
I. If a parallelogram is inscribed in a circle, it must be a rectangle.
2. Doubling the minor arc of a circle will double the chord of the arc.
3. On a sphere, exactly two circles can be drawn through two points which are not ends of a diameter.
4. An equilateral polygon inscribed in a circle must be equiangular.
5. A radius of a circle is a chord of the circle.
6. If an inscribed angle and a central angle subtend the same arc, the measure of the inscribed angle is twice the measure of the central angle.
7. A straight line can intersect a circle in three points.
8. A rectangle circumscribed about a circle must be a square.
9. The angle formed by two chords intersecting in a circle equal in degrees to half the difference of the measures of the intercepted arcs.
10. A trapezoid inscribed in a circle must be isosceles.
] I. All the points of an inscribed polygon are on the circle.
12. Angles inscribed in the same arc are supplementary.
13. A line perpendicular to a radius is tangent to the circle.
14. The angle formed by a tangent and a chord of a circle is equal in degrees to one-half the measure of the intercepted arc.
15. The line joining the midpoint of an arc and the midpoint of its chord is perpendicular to the chord.
16. fhe angle bisectors of a triangle meet in a point thatisequictisranrfrofh the three sides of the triangle.
17. Two arcs are congruent if they have equal lengths.
18. If two congruent chords intersect within a circle, the measurements of the segments of one chord respectively equal the measurements of the segments of the other.
19. The line segment joining two points on a circle is a secant.
20. An angle inscribed in an arc less than a semicircle must be acute.
21. The angle formed by a secant and a tangeilt intersecting outside a circle is measured by half the sum of the measures of the intercepted arcs.
22. If two chords of a circle are perpendicular to a third chord at its endpoints, they are congruent.
~3. An acute angle will intercept an arc whose measure is less than 90.
2:. A ch~rd of a ~ircle is a.diameter.
26. The mtersectwn of a lme and a Circle may be an empty. set.
27. Spheres are congruent iff they have congruent diameters.
. If.a plane and a sphere have more than one point in common, these POints lie on a circle.
243
Test 3
PROBLEMS
Find the number of degrees in La, Lf3, and s in each of the following:
Prob.1.
Prob.3. 100°
105°
Prob.5.
35°
244
Prob.2.
Prob.4.
Prob.6.
181
Proportion - Similar Polygons
8.1. Ratio. The communication of ideas today is often based upon com- paring numbers and quantities. When you describe a person as being 6 feet tall, you are comparing his height to that of a smaller unit, called the foot.
When a person describes a commodity as being expensive, he is referring to the cost of this commodity as compared to that of other similar or different commodities. If you say that the dimensions of your living room are 18 by 24 feet, a person can judge the general shape of the room by comparing the dimensions. When the taxpayer is told that his city government is spending c!~ per cent of each tax dollar for education purposes, he knows that 42 cents Ollt or every 100 cents are used for this purpose.
The chemist and the physicist continually compare measured quantities in the laboratory. The housewife is comparing when measuring quantities of ingredients for baking. The architect with his scale drawings and the machine draftsman with his working drawings are comparing lengths oflines in the drawings with the actual corresponding lengths in the finished product.
Definition: The ratio of one quantity to another like quantity is the quo- tien t of the first divided by the second.
It is important for the student to understand that a ratio is a quotient of measures of like quantities. The ratio of the measure of a line segment to that of an angle has no meaning; they are not quantities of the same kind. We can find the ratio of the measure of one line segment to the measure of a second hne segment or the ratio of the measure of one angle to the measure of a second angle. However, no matter what unit oflength is used for measuring two segments, the ratio of their measures is the same number as long as the same unit is used for each. I n like manner, the ratio of the measures of two 245
46
FUNDAMENTALS OF COLLEGE GEOMETRYngles does not depend upon the unit of measure, so long as the same unit is sed for both angles. The measurements must be expressed in the same.
nits.
A ratio is a fraction and all the rules governing a fraction apply to ratios.
\Ie write a ratio either with a fraction bar, a solidus, division sign, or with the ymbol : (which is read "is to"). Thus the ratio of 3 to 4 is to 3/4, 3 --;-4, or 3: 4., the 3 and 4 are called terms of the ratio.
The ratio of 2 yards to 5 feet is 6/5. The ratio of three right angles to two' straight angles is found by expressing both angles in terms of a common unit
(such as a right angle). The ratio then becomes 3/4. \'
A ratio is always an abstract number; i.e., it has no units. It is a number-) considered apart from the measured units from which it came. Thus in Fig.:J 8.1, the ratio of the width to the length is 15 to 24 or 5: 8. Note this does no~-1
24" 1I
15"
Fig.B.l.
----
Exercises
1. Express in lowest terms the following ratios:
(a) 8 to 12.
(b) 15 to 9.
(c) i~ . (d) 2x to 3x.
( e) l~Sto t.
2. What is the ratio of:
(a) 1 right L to 1 straight L?
--