Page 16 - Standard Handbook Of Petroleum & Natural Gas Engineering

Geometry 5

L = q/l80*nr

Some of the properties of arcs are defined by the following theorems:
1. In congruent circles, if two chords are congruent, so are the corresponding
minor arcs.
2. Tangent-Secant Theorem-If given an angle with its vertex on a circle, formed
by a secant ray and a tangent ray, then the measure of the angle is half
the measure of the intercepted arc.
3. Two-Tangent Power Theorem-The two tangent segments to a circle from an
exterior point are congruent and determine congruent angles with the
segment from the exterior point to the center of the circle.
4. Two-Secant Power Theorem-If given a circle C and an exterior point Q, let
L, be a secant line through Q, intersecting C at points R and S, and let L,
be another secant line through Q, intersecting C at U and T, then

QR QS = QU QT
5. Tangent-Secant Power Theorem-If given a tangent segment QT to a circle
and a secant line through Q, intersecting the circle at R and S, then
QR QS = QT2
- -
6. Two-Chord Power Theorem-If RS and TU are chords of the same circle,
intersecting at Q, then
QR QS = QU QT

Concurrency
Two or more lines are concurrent if there is a single point which lies on all
of them. The three altitudes of a triangle (if taken as lines, not segments) are
always concurrent, and their point of concurrency is called the orthocenter. The
angle bisectors of a triangle are concurrent at a point equidistant from their
sides, and the medians are concurrent two thirds of the way along each median
from the vertex to the opposite side. The point of concurrency of the medians
is the centroid.
Similarity
Two figures with straight sides are similar if corresponding angles are con-
gruent and the lengths of corresponding sides are in the same ratio. A line
parallel to one side of a triangle divides the other two sides in proportion,
producing a second triangle similar to the original one.

Prisms and Pyramids
A prism is a three dimensional figure whose bases are any congruent and
parallel polygons and whose sides are parallelograms. A pyramid is a solid with
one base consisting of any polygon and with triangular sides meeting at a point
in a plane parallel to the base.
Prisms and pyramids are described by their bases: a triangular prism has a
triangular base, a parallelpiped is a prism whose base is a parallelogram and a