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

ISBN 0-471-15045-2

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Isosceles triangles frequently occur in circle geometry. In figure b, point O is the center of the circle. Triangle OAB is isosceles because OA = OB, since both lengths are radii of the circle. Therefore, angle A = angle B.

Example. Figure c shows a garden roller being pulled along a lawn. The handle makes an angle of 100Â° with the vertical. Find the size of the angle marked x in the figure.

Solution. Triangle OBC is the important part, and is drawn slightly larger on the right-hand side of the figure. Write

Triangle OBC is isosceles OA = OB, since both lengths are radii of the circle

Angle OCB = angle OBC Base angles of an isosceles triangle are equal

B

c

c

(c)

Angle OCB = 40Â° x = 50Â°

Sum of the angles of a triangle = 180Â° Radius is perpendicular to the tangent

References: Angle Sum of a Triangle, Angles on the Same Arc, Circle Geometry Theorems, Symmetry, Tangent and Radius Theorem.

K

KILO

Kilo is a prefix meaning 1000. For example, 1 kilometer means 1000 meters. The abbreviation for kilometer is km, where k stands for kilo and m stands for meter. The metric units are mainly based on multiplying some quantity by 1000 (so the unit is preceded by the prefix kilo) or dividing some quantity by 1000 (so the unit is preceded by the prefix milli).

References: CGS System of Units, Metric Units, SI Units.

KILOGRAM

One kilogram means 1000 grams. Kilogram is abbreviated kg, where k stands for kilo and g stands for gram.

References: CGS System of Units, Kilo, Metric Units, SI Units.

KILOLITER

One kiloliter means 1000 liters. Kiloliter is abbreviated kl, where k stands for kilo and 1 stands for liter.

References: CGS System of Units, Kilo, Metric Units, SI Units.

KITE

A kite is a convex quadrilateral. Convex means that all its angles are less than 180Â°, and quadrilateral means it has four sides (see figure). A kite has one axis of symmetry, and two pairs of adjacent sides are of equal length. Some kites that people fly at the end of a string on a windy day are kite shaped.

269

270 KÃ–NIGSBERG BRIDGE PROBLEM

m

A re-entrant quadrilateral with one axis of symmetry and two pairs of adjacent sides of equal length is called an arrowhead (see far right of figure). An arrowhead has one reflex angle.

References: Adjacent Angles, Convex, Quadrilateral, Reflex Angle, Symmetry.

KÃ–NIGSBERG BRIDGE PROBLEM

This is a problem from the history books, which is still popular today. In the year 1736 Leonhard Euler solved a problem which interested the residents of the town of KÃ¶nigsberg, in Prussia. The town was close to the river Pregel, which divided and flowed around an island. There were seven bridges over the river as shown in the figure. The island is drawn shaded.

Problem. Start at any point, say point A, and walk over every bridge only once and return to your starting point, without getting your feet weti

Solution. Euler reduced the bridge problem to a geometry problem. The process of changing a real-life situation into an abstract mathematical problem is called modeling. He constructed a network in which lines represented the routes joining the positions of A, B, C, and D by passing over the bridges, and obtained the right-hand drawing in the figure. In order to understand Eulerâ€™s explanation, you will need some knowledge of networks, which is found under that entry. The problem now becomes: Can a network be drawn, without taking the pen from the paper, by starting at point A, traveling over every line once, and finishing up back at the point A? You can pass through a node any number of times, but you cannot redraw a line. In other words, is the figure traversablel

B

D

KÃ–NIGSBERG BRIDGE PROBLEM 271

A network is traversable if it can be drawn without taking the pen from the paper and traveling over every arc just once. You can pass through a node any number of times, but you cannot redraw a line. Euler discovered the following rale for a network to be traversable, whether you return to your starting point or not:

To be traversable a network must have either two odd nodes or it must have no odd nodes. It may have any number of even nodes.

In Eulerâ€™s figure representing the bridge problem, the order of the node A is 3, that of node B is 3, that of node C is 5, and that of node D is 3.

This network has four odd nodes, and is therefore not traversable. Eulerâ€™s conclusion was that the route round the town crossing each of the seven bridges only once and returning to the starting point was impossible.

References: Networks, Traversable Networks.

L

LAWS OF INDICES

Reference: Exponent.

LENGTH

The length of a line segment, which is part of a straight line, is the shortest distance between its end points. A line has one dimension, whereas area is a quantity involving the two dimensions of length and width, and length is usually taken to be greater than width. The principal unit we use for measuring length is the meter. The length of a straight-line segment is measured using a ruler or tape measure. The length of a curved line can be measured by laying a piece of string along it and then measuring the length of string with a ruler, or using a flexible tape measure.

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