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Solution. The method will be a geometrical construction on the map. If the toilet block (T) is to be equidistant from each of the three points A, M, and G, then its position will be at the circumcenter of the circle that passes through the triangle AMG. The steps of the construction refer to figure c. Use a ruler to join the three points G, A, M. With ruler and compasses, construct the perpendicular bisectors of two sides of the triangle. The point where they intersect is the center of the circumcircle of triangle GAM. The method is explained in the entry Center of a Circle, method 2. The center of the circle T is the place where the toilet block should be erected, because T is equidistant from the points G, A, and M.
Reference: Center of a Circle.
CLASS INTERVAL 79
The circumference of a circle refers to either the curved boundary line of the circle or the length of the boundary of the circle. Circumference is a word that also applies to other closed geometric figures. The circumference (C) of a circle can be calculated using the formula C = ttD, where D is the diameter of the circle. An example of this calculation is given in the entry Circle.
References: Circle, Revolution.
This means to draw a circle, or any closed curve, around the outside of a polygon so that the circle passes through all the vertices of the polygon.
Figure a shows how a square peg fits snugly into a round hole. The round hole circumscribes the square peg. The sides of the square are chords of the circle. Inscribe means to draw a circle inside a polygon so that all the sides of the polygon just touch the circle. Figure b shows the view from the top of a round cake in a square box. The circle is inscribed in the square. The sides of the square are tangents to the circle.
References: Chord, Polygon, Tangent.
The following grouped frequency table shows the marks out of 100 obtained by 100 students who took a mathematics examination:
Mark 0-9 10-19 20-29 30-39 40-49 50-59 60-69 70-79 80-89 90-100
Frequency 1 3 12 15 20 23 12 8 4 2
The marks from 0 to 100 are divided up into the intervals 0-9, 10-19, 20-29, etc., in order to make the data more manageable. These subdivisions are called class intervals. In the table above there are 10 class intervals. Class limits are the greatest and least values in an interval. In the table, the class limits are 0 and 9,10 and 19, 20 and 29, etc.
Reference: Arithmetic Mean.
Reference: Class Interval.
The coefficient of a variable in an algebraic term is the number that multiplies the variable. The coefficient of the variable x in the term 3x is 3, since 3 multiplies the variable x. Additional examples are
♦ The coefficient of x2y in the term —Sx1 y is —5.
♦ The coefficient of xy in the term 3x + 5.x y is 5, and the coefficient of x is 3.
♦ The coefficient of xy in the term 2 + xy is 1, because 1 xy is written as xy.
Binomial coefficients were used by Blaise Pascal (1623-1662) to expand brackets of the type (a + b)n, where n is a whole number. At the time he was trying to solve problems in probability. Pascal expanded these brackets by multiplying the brackets together for n = 1, 2,3,4,..., and noticed a pattern in the coefficients (see table). Using that pattern, he was able to expand brackets of the type (a + b)n, where n is a whole number.
Brackets Expansion Coefficients
(a+h)° 1 1
(a + b)1 a + b 1 1
(a + b)2 a2 +2 ab+lf 1 2 1
(a + bf a3 + 3 a2b+ 3aif + if 13 3 1
(a + b)4 a4 + 4a3 b+ 6 a2 if + 4 atf + b4 1 4 6 4 1
Arranging the coefficients in the shape of an isosceles triangle, Pascal noticed a pattern, which is explained below (see figure). In the expansion of (a + b)n, the algebraic terms in a and b can be easily worked out. The power of a reduces by one each time, from n to zero. The power of b increases by one each time, from zero to n.
COLLECTING DATA 81
1 2 1 13 3 1
1 4 6 4 1
1 5 10 10 5 1
The coefficients can be worked out by continuing Pascal’s triangle, which is the name given to this table of coefficients. Each number, other than 1, is the sum of the two numbers standing above it to the left and to the right. For example, 10 = 4 + 6, and 4 = 3 + 1. The table can be continued further as needed.
Example. Expand the bracket (a + b)5.
Solution. The line of coefficients, from Pascal’s triangle, that is to be used is 1,5, 10,10, 5, 1. In the expansion of (a + b)5 the terms in a start at a5, and the next terms are a4, a3, a2, a1, and a°, which is 1. The terms in b start at b°, which is 1, and the next terms are bl, b2, b3, b4, and b5. Combining the coefficients, the terms in a and the terms in b, and remembering that a1 = a, we obtain the expansion
(a + b)5 = a5 + 5 a4b + 10 a3b2 + 10 a2b3 + Sab4 + b5
References: Brackets, Expanding Brackets, Pascal’s Triangle, Powers, Variable, Whole Numbers.