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3. We know from geography that lines drawn on a map from right to left are east-west lines. East is to the right; west is to the left. Make a drawing like Fig. 13, and through each of the points A, B, C, D, E, draw the east-west line.

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4. In the kind of drawing or diagram which we call a map, upward means north, or toward the top of the page, downward means south, or toward the bottom of the page. Make a drawing like Fig. 14, and through each of the points A, B, C, D, E, draw the north-south line.

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5. To draw straight lines it is necessary to have a good ruler. Test the straightness of your ruler as follows:

On the blackboard or on a sheet of paper mark two points, A and B.

Placing the ruler upon the blackboard or the paper so that the edge passes through A and B, draw a line through A and B.

Then place the ruler on the opposite side of the line AB, making the edge again pass through A and B, and draw a line through A and B.

If the edge of the ruler is straight and if the drawing is well made, the second line should fall exactly on the first. The two lines are then said to coincide (fall together).

This exercise illustrates the fact that through two given points only one straight line can be drawn.

5. Line segment. In the preceding paragraphs the word “line” has been used without considering length. A geometric line is unlimited in length.

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A limited portion of a line, i.e., one which is bounded by two points like A and B (Fig. 15), is a line segment. Note that the difference between line segment AB and line AB is that the first is bounded by two points, while the second extends indefinitely.

6. How to measure length. Lengths may be measured with various instruments. To measure the length of a line segment as AB (Fig. 16) with a ruler,

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place the edge, which is marked off into equal parts, along AB with the zero mark directly under A.

Then read off the mark on the ruler which is directly under B. If the ruler is graduated in inches the reading gives the number of inches contained in AB. This number is the length of AB, and the inch is the unit segment. Thus, to measure a line segment is to determine how many times it contains another segment, called the unit segment.

EXERCISES

1. Look again at Fig. 16 and tell the length of AB.

2. Draw a line segment and find the length by measuring with a ruler as you were shown in $6.

In this exercise some of you have discovered that when we are measuring segments, we cannot always find the exact length, because the end point of the segment does not in every case fall exactly over a mark of the ruler. The length then has to be estimated. Thus, in Fig. 17 the length of AB is greater than 216 and less than 21. It seems to be nearest to 216. The length is said to be 216 approximately, or 21e to the nearest sixteenth of an inch.

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3. State how one measures a line segment with a ruler.

4. Draw a line segment and find the length to the nearest sixteenth of an inch. Write the result in the form used in the final statement in Exercise 2.

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Add the lengths of the three segments and divide the sum by 3.

6. Make a drawing like Fig. 19. Measure each of the three segments. Find the sum and divide it by 3. Arrange the work as in Exercise 5.

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7. Draw five R

segments and name them AB, CD, EF, GH, IK. Measure each to the nearest sixteenth of an inch.

Find the sum and Q

divide the sum

by 5. Fig. 19

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drawing like the rectangle in Fig. 9 and measure each side to the nearest sixteenth of an inch.

9. Draw a square (Fig. 9) and measure each side to the nearest sixteenth of an inch.

7. Arithmetical average. In a problem in measuring, the results of all pupils in a class usually do not agree exactly. Some pupils work more accurately than others. If we compare classes, we find that some classes work more accurately than others. If we add the results when all pupils in a class have measured a line segment, and then divide the sum by the number of pupils in a class, we obtain the average length, or the arithmetical average. Likewise, the arithmetical average of several segments is found by adding the lengths of all the segments and dividing the sum by the number of segments. The average of the results obtained by all the pupils in a problem in measurement is used to tell how accurately the class can measure. We may also find the separate averages for the boys and the girls and, by comparing each of these averages with the correct measure, we can compare the work of the boys with that of the girls.

The following example shows how to find the average of several numbers.

Find the average of 63, 41, 58.

Solution: Changing all fractions to the same denominator, we have 63+41+5=612 +412+512

= 1611 To find the average,

Computation:

16 divide 1612 by 3.

12 Average = {X1612 = 1X1022

32 16

203

= 36

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Before adding the fractions in the preceding example we had to change them to fractions having a common denominator. The following is a simple way of finding the least common denominator of several fractions, as }, }, and .

1. Choose the largest of the denominators, i.e., 10.

2. Multiply it by 2, then by 3, etc., until a multiple of 10 is found which contains each of the denominators as a divisor. Thus, 2X10 and 3x10 contain 5 but not 8. However, 4X10 contains both 5 and 8. It is the required least common multiple of the denominators.

Similarly, to find the least common denominator of 18, 15, and swe write down the largest denominator,

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