PROPOSITIONS OF GEOMETRY Table 1e. Propositions of Geometry Machinery's Handbook, 31st Edition
65
A major arc of a circle is one that measures more than half the circumference of the circle. A minor arc measures less than half the circumference of the circle. In the figure, angle A is subtended by a minor arc, so it is an acute angle (it measures less than 90 degrees). Angle B is subtended by a major arc, so it is an obtuse angle (it measures more than 90 degrees). An angle subtended by a circle’s diameter is a right angle; the arc described by the diameter is a semicircle. Referring to the figure, right angle A is subtended by diameter d . Angle C = 90°
A
B
In the figure, the product of line segment lengths formed by intersecting chords in a circle are equal. Thus: ab = cd
c
b d
a
When two lines are drawn from a point outside a circle, one tangent and one through the circle intersecting it at two points, the line segment lengths are such that the square of the tangent segment is equal to the product of the segments formed by the other line. Thus: a 2 bc =
a
c
b
b
a
Arc lengths of a circle are proportional to the corresponding central angle measures. Thus: A : B a : b =
B A
b
a
The lengths of circular arcs having the same central angle are proportional to the lengths of the radii. Thus, if A = B , then a / b = r / R
B
A r
R
The ratio of the circumferences of two circles is proportional to the ratio of their radii. c : C :: r : R so c / C = r / R The ratio of the areas of two circles is proportional to the ratio of the squares of their radii. a : A :: r 2 : R 2 so a / A = r 2 / R 2
Circum. = C Area = A
Circum. = c Area = a
R
r
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