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

ISBN 0-471-15045-2

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The interquartile range = 4 and the range = 8 are calculated as before.

References: Box and Whisker Graph, Central Tendency, Cumulative Frequency Graph, Mean. INTERSECTING CHORDS

There are three geometry theorems under this heading about the lengths of chords that intersect each other under the following circumstances:

♦ Inside a circle

♦ Outside a circle

♦ When one of the chords is a tangent to the circle

Theorem 1. When two chords AB and CD intersect inside a circle at a point X (see figure a), the lengths of the chords are related as

AX x XB = CX x XD

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Example 1. A knife and a fork are arranged on a dinner plate so that their ends just reach to the edge of the plate, as shown in figure b. The measurements on the figure are in centimeters. Find the length of the fork.

INTERSECTING CHORDS 261

Solution. Let the length of the prongs of the fork be x cm. Write

16 x 6 = 12 x jc Intersecting chords inside a circle 96 = I2x

x = 8 cm Dividing both sides of the equation by 12 The length of the fork is 20 cm, because 12 + x =20, when x = 8.

Theorem 2. For two chords AB and CD that intersect outside a circle at a point X (see figure c; the lines AX and CX are called secants), the lengths of the lines are related as

AX x XB = CX x XD

A

Example 2. In figure c, BX = 3 cm, AB = 4 cm, and XD = 3.5 cm; find the length of CD.

Solution. Write

AX x XB = CX x XD Intersecting chords outside the circle

7 x 3 = CX x 3.5 AX = AB + BX

21 = CX x 3.5

CX = 6 Dividing both sides of the equation by 3.5

CD = 2.5 cm CD = CX — XD

Theorem 3. In the special case of chords that intersect outside the circle and one of the chords is a tangent to the circle (see figure d), the lengths of a secant and a tangent to the circle are related as

AX x XB = XC2

262 INTERVAL

Example. A watchtower which looks out to sea is of height 100 meters above sea level. The watchman, Bill, is on duty. If the diameter of the earth is 12,800 km, how far can Bill see to the horizon?

Solution. Figure e shows a cross section of the earth. The tower is represented by the line XB. Bill can see a distance that is represented by the length of the tangent XC. In order to maintain the same units throughout the calculation, the height of the tower must be in kilometers.

The height of the tower is given by

XB = 0.1 km Divide meters by 1000 to change into kilometers

Now,

AX x XB = XC2 Tangent secant theorem

12,800.1 x 0.1 = XC2 XB = 0.1 andX4 = 12,800 + 0.1

XC2 = 1280.01

XC = 35.78 (to 2 dp) Taking square roots The distance Bill can see to the horizon is 35.78 km.

References: Chord, Circle Geometry Theorems, Quadratic Formula, Secant, Tangent. INTERVAL

An interval is the set of values between two end points of a line. If the interval includes the end points, it is a closed interval; if it does not include the end points it is an open interval. The interval drawn in figure a is the set of all numbers between 3 and 7, and including 3 and 7. The solid circles at each end of the interval means that those

INVERSE OPERATION 263

numbers are included. This interval can be written as 3 < x < 7 if x is the variable, or as [3, 7]. This is a closed interval.

•--------------• o o

(a) (b)

The interval drawn in figure b is the set of all numbers between 3 and 7, but excluding 3 and 7. The open circles at each end of the interval means that those numbers are not included. This interval can be written as 3 < x < 7 if x is the variable, or as (3, 7). This is an open interval.

An interval can be half-closed, which means it is closed at one end and open at the other end. For example, 3 < x < 7 can be written as [3, 7).

Reference: Greater Than.

INVARIANT POINTS

These are points that are not altered by a transformation. They can be described as fixed points under a transformation. For more information see the entries on transformations: Enlargement, Reflection, Rotation, Translation.

Reference: Transformation Geometry.

INVERSE OPERATION

The inverse of an operation is another operation which “undoes” the first operation. This is explained by means of the following example.

Example 1. Think of number: say, 7. The operation on this number is to multiply it by 5: 7 x 5 = 35. To undo the operation “multiply by 5” is to do an operation on 35 to take it back to the starting number 7. This operation is “divide by 5.” So the inverse operation of “multiply by 5” is “divide by 5,” which is the same as “multiply by |.”

In general terms, the inverse operation of multiplying by x is dividing by x or multiplying by 1/x. Similarly, the inverse operation of adding x is subtracting x or adding —x. The inverse operation of squaring is square rooting. The inverse operation in trigonometry is illustrated with the following example.

Example 2. Using a calculator, we can see that sin 30° = 0.5. To undo this operation, we use the inverse sine in the following way:

sin HO.5) = 30° Check this on your calculator

264 INVERSE RELATIONS

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