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To represent graphically the facts stated in this table lay off, to a convenient scale, the values of d horizontally and the values of c vertically (Fig. 264).

From the graph find c when d=13; 2); 63.

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137. Direct variation. To the table ($136), add a

third line giving the ratio , for each pair of values of

d c and d.

Notice that as we let d vary (change) from 1 to 8, also varies. A change in d affects the values of c. For example, if d is doubled, c is also doubled. However,

с

the ratio remains always the same.

This law of

variation is known as direct variation. When we say c varies directly as d, we mean that c and d vary in

с

such a way that the ratio ; remains constant (the same).

d The graph (Fig. 264) represents geometrically the relationship between c and d which is expressed algebraically by the formula c=ad.

This type of variation is very common. We meet it in a variety of problems, some of which will be studied in Chapter VIII.

138. What every pupil should know and be able to do.

All pupils should know how to make the following constructions with ruler and compass:

1. To draw angles of 60°, 30°, 15°, 120°, 90°, 45°, 135o. 2. To bisect an angle. 3. To bisect a segment.

4. To draw a perpendicular to a line at a point on the line; from a point not on the line.

5. To draw a line parallel to a given line. 6. To draw an angle equal to a given angle. 7. To divide a circle into 4 equal arcs; 6 equal arcs.

8. To draw an inscribed regular hexagon; equilateral triangle; square.

9. To make copies of simple designs. 10. To solve problems by means of the circumference formula. 11. To represent graphically the relation c=ad. 12. The following theorems should be known:

a. Two triangles are congruent if three sides of one are respectively equal to three sides of the other.

b. The circumference of a circle is equal to a times the diameter, or 21 times the radius, i.e., c=id, or c=27r.

139. Typical problems and exercises. Pupils should be able to give correct answers to the following questions and problems:

Solve by means of equations:

1. The circumference of a circle is 48 feet. Find the radius.

2. The diameter of a circle is 12 feet. Find the circumference.

3. The diameter of a wheel is 3 feet. Find the number of revolutions it makes in passing over a distance of one mile.

4. Make a graph of the equation crad.

5. Write a paper on one of the following topics:

a. The use of the circle in construction exercises. b. The history of finding the value of n.

CHAPTER VIII

FORMULAS AND EQUATIONS

LITERAL NUMBERS WHICH CHANGE IN VALUE

140. Equations studied in preceding chapters. In the preceding chapters we have used the equation as a tool for solving problems. All of the equations, so far, have been of a simple type and easily solved. Thus, in studying perimeters, we found equations of the form 120=6s. In studying triangles we used equations like 4x+2x+5x=180 to express the sum of the angles. The equation x+3x =90 may mean that two angles are complementary, and m+5m=180 may mean that two angles are supplementary. The acute angles of a right triangle satisfy relations like a+60=90. The circumference of a circle is found by means of the formula c=ud. Similar triangles lead to equations like

8 5 15°

All these illustrations show that one cannot go very far in the study of mathematics without a knowledge of algebra (in which letters are often used for numbers), in particular of equations. However, all the equations above can be simplified, as may be seen from the table below.

In each case show how the first form is reduced to the simplified form.

X

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In the simplified form all the equations above are of one and the same type. Each may be solved by dividing both members of the equation by the coefficient of the unknown number. It is the aim of this chapter to make us more familiar with equations of this form by bringing in problems from fields other than mathematics; and to extend our knowledge of equations to others which reduce to the same type form, such as 4x+8= 20, or 3x - 2 = 13.

141. The law of direct variation. We have seen ($137) that in the formula c=id the value of c depends upon the value of d. For every value of d, there is a corresponding value of c. If d is made to vary (change)

c raries also, but the ratio ä remains the same. Stating

this relation between c and d in words, we say that the circumference varies directly as the diameter, or that it is directly proportional to the diameter.

The following is an example of direct variation. If a yard of cloth sells at $3, 2 yards sell at $6, 3 yards

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