6. Draw a segment equal to AB, Figure 2, by using the

compasses.

SOLUTION. From a point M draw a line MX at least as long as AB. Open the compasses the distance AB. Mark off from M on MX the distance MN equal

M

to AB. Then MN is the required line segment.

FIG. 4.

N X

7. Draw two line segments of different lengths. By using the compasses make two segments equal to these segments.

8. Draw a line segment. Draw a segment equal to it, first by using the ruler only, and then by using ruler and compasses. Which method is probably more accurate?

9. How many straight lines can be drawn through a given point P? How many straight lines can be drawn so as to pass through both of two given points A and B? Can you always draw a straight line that will pass through any three given points X, Y, and Z?.

NOTE. To do the work of the remaining chapters in this book each pupil should be supplied with the following materials: a hard pencil and a medium pencil, both well sharpened; six-inch celluloid ruler and foot-ruler; compasses, protractor, and eraser; loose-leaf notebook, with unruled paper.

· Suggestions for the teacher. Constructions should be made in class at first.

Insist upon neatness and as high a degree of accuracy as may be expected of pupils of this grade.

Do not accept careless and inaccurate work. Have the constructions repeated until they are done satisfactorily.

The constructions should be made in permanent notebooks. The figures should be made in fine lines with a hard pencil. Each construction in the notebook should be numbered for convenience in grading.

Much of the examination of notebooks may be made during the recitation period while the figures are being drawn.

The room should be provided with a blackboard protractor.

10. The point A is called a corner or vertex of the cube. How many vertices has the cube?

11. The line AB is called an edge of the cube. How many edges has a cube?

FIG. 5.

12. What is the sum of the edges of a cube if one edge is 4 in.? If one edge is e units. long? Make a formula for finding the sum of the edges of a cube.

13. Find the length of one edge of a cube if the sum of the edges is 40 in. Make a formula for finding one edge, e, when the sum of the edges, s, is known.

14. A box is 4 ft. long, 18 in. high, and 23 ft. wide. When the box is nailed up for shipping a metal strip is nailed along each edge to make the box stronger. How many feet of such metal strip will be required to strengthen 100 boxes in this way?

15. The point O is called the vertex of the pyramid. The line OA is called a lateral edge of the pyramid. ABCD is the base of the pyramid. How many edges has this pyramid? If the base is a 6-inch square, and each lateral edge is 8 in., what is the sum of the edges of this pyramid? If. the length of each edge of the base is e and the length of each lateral edge is 7, make a formula for finding the sum of the edges.

D

A

FIG. 6.

C

B

57. Finding the sum and the difference of line segments. In making drawings it is often necessary to draw a line equal to the combined lengths of two or more lines. The sum and also the difference of line segments may be found by using the compasses and without measuring the segments.

Exercise 61

1. Find the sum of the segments, a, b, and c.

SOLUTION. Draw a line OX of indefinite length from the point O. Then open the compasses a distance a, and lay off from the point O a segment equal to a. Then lay off b and c in a similar way, as in the figure. The segment OM is the sum of a, b, and c. We may write OM=a+b+c.

2. Draw two line segments and find their sum.

3. Draw a triangle. Draw a line equal to the sum of its sides.

4. Draw a line segment equal to the difference of the given segments a and b.

SOLUTION. From a point O draw a line at least as long as a. Lay off from O a segment ON equal to a. Then from N lay off toward O a segment NM equal to b. The segment OM is the difference between a and b. We may write

OM = a -b.

5. Draw three line segments a, b, and c.

of a and b take c.

From the sum

6. Draw a triangle. Subtract the longest side from the sum of the other two sides.

58. Accuracy in measuring. Let each of three pupils measure the length of the schoolroom with a foot-ruler and compare the results.

In applying a ruler you place one end on a mark previously made and mark the position of the other end. Errors are probable in doing both these things. The ruler may be a little too long or a little too short. For such reasons as these no measurement can be known to be entirely accurate. It is very important for a pupil to know how to make his errors in measuring as small as possible, and to know what degree of accuracy may be expected under given conditions.

Exercise 62

1. In measuring with a foot-ruler you place one end on a mark previously made and mark the position of the other end. In doing either of these things can you be sure that you will not make an error of in.? Of in.? Of in.? Of in.? Of in.? Of .01 in.? Of .001 in.? We will suppose that the divisions are accurately marked on your ruler, that it has square ends, and that you have a well-sharpened pencil.

2. A boy's height is marked on the wall. It is measured by each of 4 pupils. The results are 4 ft. 3 in., 4 ft. 3 in., 4 ft. 3 in., and 4 ft. 3 in. What is the greatest difference between any two measurements? Do you know the boy's height? Are these results probably accurate within 1 in.? Within in.? Within in.?

3. Let each pupil measure the length of this page as accurately as possible. Write the results on the blackboard. Do you know the length of the page within 1 in.? Within .01 in.?

4. Let each pupil measure the length of the second line of print on this page. Compare the results. What is the greatest difference between any two?

5. Mark the height of a pupil on the blackboard. Let each of 4 pupils measure it. What is the greatest difference between any two measurements? How accurately do you know this pupil's height, that is, to what fraction of an inch?

6. If your height is marked on the blackboard, about how accurately can you measure it with a foot-ruler?

7. About how accurately can you measure the length of your schoolroom with a foot-ruler? Let each of 6 pupils measure it. Are you sure of the length within 1 ft.? Within 1 in.? Within in.? Let each of 6 pupils measure it with a yardstick. What degree of accuracy do you think you now have?

8. You have probably concluded that measuring cannot be expected to be exact. When we say that we have measured a board and that it is 16 ft. long, we do not mean that we know that the length is exactly 16 ft., but that the length differs from 16 ft. by a small amount. This amount will depend upon care in measuring, accuracy of the measuring instruments, and the length of the distance measured.

If a boy does not make an error of more than in. in each application of a foot-ruler, what is the largest error that he can make in measuring a line that is between 11 and 12 ft. long? If he does not make an error of more than in. in each application of a yardstick, what is the largest error that he can make in measuring the same line with a yardstick?

9. John is making some bookshelves. The shelves are between 3 ft. and 4 ft. long, and he is measuring with a footruler. To have a fairly good job he must not make an error of more than in. in the length of a shelf. Can you measure as accurately as that?

10. Suppose that an error of 3 in. is made in measuring a length known to be 20 ft. What is the per cent of error? If the same per cent of error is made in measuring a mile, how many feet does it amount to?