# The A to Z of mathematics a basic guide - Sidebotham T.H.

ISBN 0-471-15045-2

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Substituting —2 for x, 3 for y, and 0 for z

13

~cT

= undefined The answer is undefined; we cannot divide a number by zero

References: Algebra, Equations, Formula, Graphs, Table of Values, Variable.

SUBTEND

Subtend is a word used in circle geometry theorems. A subtended angle is an angle that “stands” on an arc of a circle. In the figure the angle x is subtended at a point P

SYMMETRY 431

by the arc AB of the circle. The arc AB referred to is highlighted in the figure. The angle x may be subtended at a point on the circumference of the circle or at its center.

Angle x is subtended at the Angle x is subtended at the

circumference by AB center by AB

References: Arc, Circle, Circle Geometry Theorems, Circumference.

SUPPLEMENTARY ANGLES

Reference: Complementary Angles.

SURFACE

A surface is the complete boundary of a solid. A well-known solid is a cube, which has six square faces, and these six faces make up the surface of the cube.

References: Cube, Solids, Surface Area.

SURFACE AREA

The surface area of a solid is the total area of its surface. To find the surface area of a particular solid, see the entry under the name of the solid.

Reference: Area.

SURVEY

References: Census, Questionnaire.

SYMMETRY

References: Axis of Symmetry, Order of Rotational Symmetry.

432 SYSTÈME INTERNATIONAL D’UNITÉS

SYSTÈME INTERNATIONAL D’UNITÉS

Abbreviated SI units. These SI units, along with some scientific units, were adopted by international agreement in the year 1960. The units in the SI system are as follows:

♦ Length: meter; abbreviation m

♦ Mass: kilogram; abbreviation kg (in everyday use, mass is sometimes called weight)

♦ Time: second; abbreviation s.

A full explanation of these metric units is given under the entry Metric Units.

References: CGS System of Units, Metric Units.

T

TABLE OF VALUES

References: Asymptote, Graphs.

TALLY CHART

Reference: Frequency.

TANGENT

Reference: Normal.

TANGENT AND RADIUS THEOREM

Suppose that T is a point on a circle with center 0, and a tangent is drawn to the circle at that point T (see figure a). If a radius of the circle is also drawn through the same point T, then the angle between the tangent and the radius is a right angle. We say that the radius is perpendicular to the tangent.

(a)

The next example involves the use of this theorem and also includes the use of another circle geometry theorem.

433

434 TANGENT TO A CIRCLE

Example. In figure b, PT and PQ are tangents to the circle, and 0 is the center of the circle. Find x and y.

Solution. Write

x = 140° Angle at the center is twice the angle at

the circumference

Angle PTO = angle PQO

= 90° Radius is perpendicular to the tangent

The figure PTOQ is a quadrilateral and the sum of its four angles is 360°:

jc + 90° + y + 90° = 360°

140° + 90° + y + 90° = 360° Substituting x = 140° y =40°

See the entry Isosceles Triangle for another example of the theorem that the radius is perpendicular to the tangent.

References: Isosceles Triangle, Tangents from a Common Point.

TANGENT OF AN ANGLE

References: Gradient, Trigonometry.

TANGENT TO A CIRCLE

References: Chord, Circle.

TANGENTS FROM A COMMON POINT

435

TANGENTS FROM A COMMON POINT

This is a geometry theorem about two tangents drawn from one point to a circle. In figure a, the point common to the two tangents PT and PQ is the point P. The theorem states that the two tangents are equal in length. In figure a, PT = PQ, and this theorem is stated as follows:

♦ Tangents from a common point are equal.

The properties of figure b are as follows: The line OP is an axis of symmetry, so the two triangles PTO mdPQO are congruent. The figure OTPQ is a cyclic quadrilateral because a pair of its opposite angles, T and P, add up to 180°. This means that angle TPQ + angle TOQ = 180°.

Example. A straight plank of wood AT rests across a circular garden roller, which stands on horizontal ground (see figure c). If the distance AB = 2 meters, what is the distance ATI

(c)

Solution. AT and AS are tangents to the circle which is the cross section of the roller. Write

AT = AB

= 2 meters Tangents from a common point are equal The distance AT is 2 meters.

References: Axis of Symmetry, Cyclic Quadrilateral, Tangent and Radius Theorem, Trigonometry.

436 TEMPERATURE

TANGRAM

A tangram is a Chinese puzzle. A square is cut up into seven shapes, as shown in figure a. The object of the puzzle is to cut out the seven shapes and rearrange all of them to form interesting figures of animals, humans, etc. Figure b shows a pelican, made up of all the seven pieces.

References: Area, Isosceles Triangle, Parallelogram, Right Angle, Square. TEMPERATURE

Temperature is a measure of how hot or cold something is. The two scales of temperature in common use are Fahrenheit (F) and Celsius (C). Celsius used to be called centigrade. The units for measuring temperature are degrees, but degrees Celsius (°C) are not the same size as degrees Fahrenheit (°F). Conversions between the two temperatures can be done using formulas, as follows.

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