Ex. 5. In chap. III, art. 12, a triangle was made from a rectangle by cutting off two corners, and these corners were experimentally fitted to the triangle to exactly cover it. Prove this result by general reasoning, and thus justify the expression found for the area of a triangle, half the product of the base by the height.

24. Cut out a paper parallelogram ABCD (Fig. 17) and

see if you can cut it in two and fit it together into a rectangle.-Cut it along a line EF perpendicular to two opposite sides, and then fit BC and AD together. You

thus have the experimental result. Can you prove it to be more than experimental, to be more accurate than errors of execution and observation make it possible for a mere experiment to show ?-Produce DC to G and AB to H. AD and BC are equal and so can be fitted together. The corresponding angles ADE and BCG are equal, so that DE will lie along CG. The angle DAF is equal to the corresponding angle CBH, so that AF will lie along BH. Thus ADEF will lie as BCKL, and the angles made by KL with CK and BL are the angles DEF and AFE, that is, right angles. Therefore the fitting together makes a rectangle EFLK.

Give an expression for the area of the parallelogram.-It is the same as of the rectangle, namely FL EF, or, since

FL is simply the length AB, it is ABX EF, the product of a side by the distance between it and the parallel side. Or, using the usual terms, it is the product of the base by the height.

Does it matter which side you choose for base?-If in the case of Fig. 17 we chose AD for base, we could not

draw a perpendicular to AD that would meet BC. DM (Fig. 18) is the best we could do.

Make the parallelo

gram in paper, cut off the piece ADM and fit it in the position DNO. What will you then do to make the figure a rectangle?-Cut off the piece OCP and fit it in the position MBQ. Prove that this gives a rectangle, and therefore, in this case also, the area is the product of the base and the height.

Ex. 6. Cut out a long thin parallelogram that will have to be cut into several pieces to fit again into a rectangle, and show how to do it. Ex. 7. Cut out a piece of paper of the form called a trapezium (Fig. 19), that is, a four-sided figure with one pair of opposite sides

parallel, and show how to cut it up and make it into a rectangle. (Take E and F the middle points of AD and BC, and cut along Ls from these points to AB.)

Also give an expression for the area of a trapezium in terms of the lengths of the parallel sides and the distance between them. Calculate the area of Fig. 19 from measurements.

25. Draw any quadrilateral ABCD, and find its area.

We join BD, draw perpendiculars from C and A on BD, and use for each triangle the expression 'half the base by the height'. Draw a number of triangles on BD as base and equal to CBD in area, as EBD, FDB, &c. (Fig. 20). What do you notice about these triangles, or what property did you use to draw them? They all have the same height, so that EFCGH is a straight line parallel to BD. Can you make a single triangle equal to the quadrilateral ABCD?- -If BH is in line with AB, the two triangles ABD and HBD form a single triangle AHD. To make DF in line with AD would also do. State formally how to make a triangle equal in area to a quadrilateral ABCD.-Through C draw a parallel to BD to meet AB produced in H. Then ADH is a triangle equal to ABCD.

26. In the same way make a quadrilateral equal in area to a given five-sided figure. Show how to make a triangle equal to any polygon, that is, to any area bounded by straight lines.

27. Draw a triangle with sides of 14, 11, 7 cm., and try to make a square equal to it. If the side of the square is x cm., what do you know of x ?—We measure the height of the triangle and so find the area to be 37.9 sq. cm. Then xxx = 37·9, and our graph of xxx (chap. III, art. 17) gives us x 6.2, and we draw the square.

This method is a trifle tedious and not very exact; we may find a better method later.

Show how to make a square equal to any polygon.

EXERCISES.

8. Bisect the angles of a rectangular sheet of paper by folding adjacent sides together, thus obtaining four crease lines running across the paper from corners to sides. At the middle of the paper a quadrilateral is thus formed by the creases. Determine its shape in any way you like, and justify your statement by a geometrical proof.

9. Use the angle-sum property to make a triangle having a = 11 cm., A = 58°, C = 65°, by calculating B and then using the values of a, B, C.

Draw any two acute angles and mark them A and C. Without measuring A and C, make a triangle having a≈11 cm. and A and C

equal to the angles you have drawn. (Find B by laying down an angle of 180° and cutting from it the angles A and C.)

10. Draw a figure ABCDE having five equal sides and five equal angles. First calculate the value in degrees of an angle.

If the sides of the figure were freely jointed at A, B, C, D, E, and if BC and AE were turned about B and A respectively until each was at right angles to AB, what would then be the values of the angles at C, D, and E?

A

B

90

45°

E

90°

11. ABC (Fig. 21) represents a triangle drawn on tracing paper, which is placed over a fixed triangle DEF so that AC is parallel to EF. Sketch a large figure (which need not be to scale), calculate the angles K, L, M, N, and write in each its value in degrees. The method of obtaining the results should be briefly indicated, and an angle referred to by one letter only.

If a pin were pushed in at C and the tracing-paper revolved, as shown by the arrow, through an angle of 10°, what would

then be the values of the angles K, L, M, N?

M

N

60

F

FIG. 21.

In each bisect

12. Make three triangles, calling each ABC. the angle A by a line AM, and from B draw BM perpendicular to AM. In each figure measure LB, LC, LMBC. Make a table like that below, and fill it up.

LB

LC

Difference of
LB and LC

LMBC

Fig. 1
Fig. 2
Fig. 3

What inference can you draw from a comparison of the last two columns ? See if you can justify your inference by geometrical reasoning.

13. ABCD is a quadrilateral figure in which AB is parallel to CD but not equal to it. Draw a figure of this kind, making AB=10 in., BC=4 in., CD=5'2 in., BD=7·1 in., and fold the page so that AB and CD fall together. Measure the part of the

crease between AD and BC and compare it with half the sum of AB and CD.

Now draw perpendiculars from C and D to AB and from A and B to the line of the crease produced, and give a geometrical proof of the relation between the lengths of AB, CD, and the

crease.

=

=

19

14. From a corner A of a rectangular sheet of paper measure off AB along one edge 24 cm., and AC along the other cm. Join BC and cut along this line. Fold the triangle to make a crease 40 through A and at right angles to BC, meeting BC in 0. Open the triangle out and fold the corners A, B, Č over to meet at 0. Open the triangle out again and, in any way you please, make a full-size drawing of it, showing the creases by dotted lines. Then answer either (a) or (b).

(a) State whether the three corners meet so that the part folded down just covers up the rest of the triangle, giving reasons why this should or should not be the case.

(b) What conclusion may be drawn about the sum of the angles of the triangle, and about the area of the triangle? Give an independent proof of one of these conclusions.

15. Imagine a triangle cut out and placed on the desk before you in any definite position. State precisely in each case the measurements you would take (a) to draw a figure of the same shape; (b) to draw a figure of the same size and shape; (c) to enable you to make a drawing of the figure in exactly the same position if the original were removed. You are to give the minimum number of measurements necessary in each case.

16. The four bars of a bicycle frame taken in order have lengths 23, 58, 62, 56 cm. Take four strips of cardboard and join them by eyelets or paper fasteners into such a frame, making each side th of its true length; the joints to be loose.

Is the form of the frame settled by the given lengths? How would you add another bar to fix the form?

An actual bicycle frame is fixed in shape with no such additional bar. How is that?

17. A four-sided field has two sides parallel and of lengths 260 and 200 yards, a third side to them and 220 yards long. The fourth side is bounded by a footpath which is to be moved so as to make the field rectangular without loss or gain of area. Draw the field, using 1 cm. to represent 20 yards, and show how the fourth side must be altered.

What is the area of the field (in acres) ?

18. ABCD is a rectilinear figure having AD parallel and unequal to BC. Show, with proof, how to make a rectangle equal to ABOD.