The

AA', BB', CC, DD, EE' are parallel straight lines. distance between each and the next being h ft., and their lengths a, b, c, d, e ft. respectively, find an expression for the area of the figure A'B'C'D'E'EDCBA. When α = 2.7, b 7.3, c = 8.2, d = 6·3, e = 2·4, h=3·1, draw the figure to scale, calculate the area, and mark it on the figure. Is the figure completely specified?

=

19. A 20-lb. steel plate originally measures 10 ft., by 4 ft. 6 in. at one end and 4 ft. 9 in. at the other, the width measurements being square to the 10 ft. edge. It is first sheared parallel as far as possible to a width of 4 ft. 7 in., next one square corner measuring 3 ft. x 2 ft. is cut off, next a circular hole 12 in. in diameter is cut in it, and, finally, a rectangular manhole measuring 23 in. by 15 in. is cut in it.

What is the weight in lb. of the remaining piece of the original plate?

20. Write down the values of 1÷x for the following values of x: 0.01, 0.1, 0.5, 1, 2, 5, 10. Show by a graph how 1÷x varies as x increases from 0.1 to 10, taking one inch as unit.

From the graph read off the value of 1÷27. Why is the graph not very useful for reading off the value of 1÷0·27 ?

21. Water has to be diverted from a river through a sixinch diameter pipe running full bore at a velocity of one foot per second to irrigate a field of 20 acres. How long will it take to deliver an inch of water over the whole area?

22. Suppose you are asked to make a copy of a four-sided figure ABCD in which AB and CD are parallel, the figure being drawn on a sheet of paper of the size that you are using. How many measurements would you have to make to reproduce the figure (1) when your copy is to have the same position on your sheet that the given figure has on its sheet; (2) when the position on the sheet does not matter? Give, in each case, one set of measurements which would be just enough, and state briefly how you would use these to draw the figure.

23. A pint of milk weighs a pounds, a pint of water weighs b pounds, and a pint of a mixture of the two weighs c pounds. What fraction (by volume) of the mixture is milk?

1.125, a = = 1·128, and c = 1.127, what fraction is milk?

CHAPTER IX

ON DRAWING TO SCALE

1. In the course of our discussions we have frequently used figures showing objects not full size, but drawn to some smaller scale. For instance, in chapter V we had a map of a piece of country, and in chapter VII we had models of cranes smaller than the real cranes, and drawings of cranes still smaller. What was the relation between lengths on the map, or the model, or the drawing, and the corresponding lengths on the original object ?-In each case all the lengths on the copy were the same fraction of the original lengths. And what was the relation between the angles of the copy and the angles of the original?— Each angle of a copy was equal to the corresponding angle of the original object. This was assumed to be so, without any special statement. We will now discuss the truth of the assumption.

2. By the help of a measuring-tape only make a plan of the playground or of a field, making every length on the plan of the actual length, that is, using a scale on

which 2 mm. represent a metre. We begin with (say) a straight piece of fence AB, measure it and lay down the corresponding length. [Figure 1 and the following figures in the text are mere sketches, and not drawn to any

particular scale.] Then we measure the distances of the tree T from A and B. Each distance gives a circle on which T lies, and their intersection gives T.

If your tape will not reach from A to T, how do you make sure of measuring in a straight line from A to T?- -One boy stands at A with one end of the tape and looks towards the tree. A second boy carries the other end towards the tree, keeping in the first boy's line of sight, and so in a straight line to the tree. When the tape is stretched, the first boy comes up, and the second goes on as before towards the tree. You rule AT on your plan with a ruler. could you test whether the edge of the ruler is straight? By looking along it. Or by ruling a line with it, and laying it in all possible ways against the line to see if it fits.

How

3. Draw another plan on a scale of 1 mm. to 1 metre (Fig. 2, sketch). How do the lengths A'B', A'T', B'T' of this figure compare with the corre

sponding lengths of Fig. 1? Compare the angles by measuring.

=

A F

B

FIG. 2.

=

In your Fig. 1 bisect AB in Cand draw CD and CE parallel to BT and AT. What equalities of lengths, angles, and areas can you now show about Fig. 1 by general reasoning? -Since DC and BT are parallel, LDCA LEBC; since AT and CE are parallel, LDAC = LECB; and we made AC CB. So that the triangles ADC and CEB are congruent, that is, will fit exactly if placed one on the other (chapter V). Since DTEC is a parallelogram, DT = CE and CD ET (chapter VIII). From the congruence of As ACD and CBE we know that CE AD and CD So that AD DT, and each is half of AT; and BE and each is half of BT. Can you show A'B'T' congruent to any triangle in Fig. 1? AB is 1/500 of the length of the fence and A'B' is 1/1000 of the same length; AT is 1/500 and A'T' 1/1000 of the distance of the tree from the point A of the fence; BT is 1/500 and B'T' 1/1000 of the distance of the tree from B. So that A'B', A'T', and B'T' are the

=

=

=

=

=

EB.

ET,

halves of AB, AT, and BT; that is, they are equal to AC, AD, and CD, and also equal to CB, CE, and EB. We know, therefore, that AA'T'B' is congruent to AADC and to ACEB (chapter V); and this gives what we wanted to know, that the angles at A and A' are equal, that the angles at B and B are equal, and that the angles at Tand T'are equal since each is equal to the angle ADC.

4. Ex. 1. How does your proof tell you that the angle at A' is equal to that at A, and not to the angle ADC or the angle ACD?

Ex. 2. Certain reasoning was quoted to show CE = DT, to show As ACD and CBE congruent, and to show ▲s A'B'T and ACD congruent. Give the reasoning in full.

Ex. 3. Show that AATB can be cut up into four triangles, each of which will exactly fit ▲ATB'.

5. Now make a plan of the fence and tree on a scale of 3 mm. to 1 metre. Call the ends of the fence on the plan X and Y, and the tree Z (Fig. 3). Divide XY into three

equal parts by cutting off XP and PQ equal to A'B'. Through P and Q draw parallels to XZ and YZ, and by the same kind of reasoning as before show the three angles of AXYZ to be equal to the three angles of AA'B'T' and to the three angles of ▲ABT.

6. Ex. 4. Compare the areas of the triangles ABT, A'B'T', and XYZ. Compare them also with the area of the triangle formed by the tree and the two ends A and B of the actual fence.

Ex. 5. Justify the following method of trisecting a straight line PQ, that is, of dividing it into three equal parts :-From P draw any line PX. Take any length on your dividers and mark off PA, AB, BC of this length along PX. Join C to Q, and

through A and B draw parallels to CQ. These parallels trisect PQ.

Ex. 6. Give some other way of trisecting the line PQ.-Measure PQ, divide the length by 3 arithmetically, and measure this distance off along PQ.

7. Show that the angles of AATB are equal to those of the triangle formed in the playground by the tree and the two ends of the fence. Suppose the length AB laid off along the fence as often as possible. Laid off 500 times it will just cover the fence. From all the points of division suppose parallels drawn, just as in Fig. 3. Then we shall have 500 triangles all congruent to ▲ATB, and a great many parallelograms, and the proof runs as before.

Show that the angles of A'T'B' and XYZ are equal to those of the triangle made by the tree and the fence. The proof as to A'T'B' does not differ from that for ATB. In the case of AXYZ if we lay off XY repeatedly along the fence, 333 lengths will not cover the fence and 334 are too long. Let us divide up ▲XYZ, as in Fig. 3, and use AXPR. XP can be laid off an exact number of times along the fence and an exact number of times along XY. And the former proof holds as between AXPR and the triangle on the playground, and also as between AXPR and AXYZ.

8. Suppose LMN (Fig. 4) a plan drawn to a scale of 1.3 mm. to 1 metre, L corresponding to A, M to B, and N to T. Show that the

LK

FIG. 4.

N

M

angles of LMN are equal to those of A'B'T'. -We want a length that can be laid off an exact number of times along LM and also along A'B'. Such a length is A'F, 1/10 of A'B', which is equal to LK, 1/13 of LM; for each of these represents the fence on a scale of 0.1 mm. to 1 metre. Draw FG || to B'T', and KU to MN. Then we show, as before, that A'G is 1/10 of A'T', FG is 1/10 of B'T', LU is 1/13 of LN, and KU is 1/13 of MN; that As A'FG and A'B'T' have their angles equal, and that As LKU and LMN have their angles equal.