Page 15 - Standard Handbook Of Petroleum & Natural Gas Engineering

4 Mathematics

triangle has no congruent sides. A set of congruent triangles can be drawn if
one set of the following is given (where S = side length and A = angle measure-
ment): SSS, SAS, AAS or ASA.
Quadrilaterals

A quadrilateral is a four-sided polygon determined by four coplanar points
(three of which are noncollinear), if the line segments thus formed intersect each
other only at their end points, forming four angles.
A trapezoid has one pair of opposite parallel sides and therefore the other
pair of opposite sides is congruent. A parallelogram has both pairs of opposite
sides congruent and parallel. The opposite angles are then congruent and
adjacent angles are supplementary. The diagonals bisect each other and are
congruent. A rhombus is a parallelogram whose four sides are congruent and
whose diagonals are perpendicular to each other.
A rectangle is a parallelogram having four right angles, therefore both pairs
of opposite sides are congruent. A rectangle whose sides are all congruent is
a square.

Circles and Spheres
If P is a point on a given plane and r is a positive number, the circle with
center P and radius r is the set of all points of the plane whose distance from
P is equal to r. The sphere with center P and radius r is the set of all points in
space whose distance from P is equal to r. Two or more circles (or spheres)
with the same P, but different values of r are concentric.
A chord of a circle (or sphere) is a line segment whose end points lie on the
circle (or sphere). A line which intersects the circle (or sphere) in two points is
a secant of the circle (or sphere). A diameter of a circle (or sphere) is a chord
containing the center and a radius is a line segment from the center to a point
on the circle (or sphere).
The intersection of a sphere with a plane through its center is called a
great circle.
A line which intersects a circle at only one point is a tangent to the circle at
that point. Every tangent is perpendicular to the radius drawn to the point of
intersection. Spheres may have tangent lines or tangent planes.
Pi (x) is the universal ratio of the circumference of any circle to its diameter
and is equivalent to 3.1415927 .... Therefore the circumference of a circle is nd
or 2nr.

Arcs of Circles
A central angle of a circle is an angle whose vertex is the center of the circle.
If P is the center and A and B are points, not on the same diameter, which lie
on C (the circle), the minor arc AB is the union of A, B, and all points on C in
the interior of <APB. The major arc is the union of A, B, and all points on C
on the exterior of <APB. A and B are the end points of the arc and P is the
center. If A and B are the end points of a diameter, the arc is a semicircle. A
sector of a circle is a region bounded by two radii and an arc of the circle.
The degree measure (m) of a minor arc is the measure of the corresponding
central angle (m of a semicircle is 180") and of a major arc 360" minus the m
of the corresponding minor arc. If an arc has a measure q and a radius r, then
its length is