4. What is the ratio of the capacities of the two models? 5. What is the ratio of the areas of the bases (ends)?

7. Model a triangular prism such as that of Fig. 1, but use for lengths 3" instead of 1" and 7" instead of 4". How does its capacity compare with that of the prism of Fig. 1, p. 168? Of Fig. 2, p. 168? Give the ratio in

each case.

8. Compare the ratios of the bases.

$103. Volume of an Oblique Prism.

The volume of a figure is the number of cubical units in the space enclosed by its bounding surfaces.

FIGURE 2

1. How many cubic inches are there in a straight pile of visiting cards 2" high, if each card is 2" ×3"?

2. How many cubic inches would there be in the pile if it were 5" high? 9" high? a in. high?

3. Push the straight pile of problem 1 over as in Fig. 1, p. 170. How many cubic inches of paper are there in this oblique pile?

FIGURE 1

4. Has the height of the pile been changed in Fig. 1? The area of the base? The volume?

5. How can the number of cubic units in a right prism be

found from the area of its base and its height? In an oblique prism?

6. Find the volumes of square prisms having edges of the following lengths:

§104. Paper-Folding. (FUNDAMENTAL GEOMETRICAL NOTIONS) For the following exercises in paper-folding any moderately thick, glazed paper will do. Tinted or colored paper, without lines, will show the creases more clearly.

It is convenient to have the paper cut into pieces, about 4" square Such paper is inexpensive and may be had of any stationery dealer.

EXERCISE I.-At a chosen point on a line, make a perpendicular to the line, by folding paper.

Fold one part of a piece of paper over upon the other and crease the paper along the fold, as at AB, by drawing the finger along the fold. Taking D to denote the chosen point, fold the paper over the point D, and bring the two parts, DA and DB, of the crease AB exactly together. Hold the paper firmly in this position and crease the paper along the line DC.

Fitting the portion of the paper between the creases DB and DC over the portion between the creases DA and DC; how do the angles, BDC and CDA, compare in size?

FIGURE 2

When two lines meet in this way making the angles at their point of meeting (intersection) equal, the lines are said to be perpendicular to each other, and each is called a perpendicular to the other.

The angles thus formed are called right angles.

EXERCISE II.-Make a perpendicular to a line at a chosen point, not on the line, by folding.

P stands for the point, not on the line shown by the crease, AB. Fold the paper back, bringing the end, A, down along the back side of the end, B. Hold carefully on the crease and slide A along until the crease PD, to be made, will go through P. Crease the paper as shown by PD. The crease PD then stands for the perpendicular to AB, through P. How does the angle PDA compare in size with the angle PDB?

A D B

FIGURE 1

EXERCISE III.-Through a given point make a line parallel to a given line by paper-folding.

FIGURE 2

The crease, AB, stands for the given line, and the dot P stands for the point the parallel to AB is to go through.

Crease a perpendicular, as PD, through P to AB as in Exercise II.

Crease a perpendicular, as PC, through P to PD, as in Exercise I. The line shown by the crease PC is parallel to the line shown by the crease AB.

EXERCISE IV.-Bisect an angle, by folding paper.

Crease two lines, as OA and OB, lying across each other. They make the angle AOB. Now fold the paper over, and bring the crease OA down on OB. Holding the creases firmly together crease the bisector OC.

1. How do the angles AOC and BOC

compare in size? Do they fit?

2. What is the ratio of angle AOC

to angle BOC? Of AOB to AOC?

B

FIGURE 3

3. How could the angle AOB be divided by creases into 4 equal parts?

4. Fold a square over a diagonal and tell how the diagonal divides the square; two of the opposite angles of the square.

5. Crease both diagonals of a square. How do they divide each other?

$105. Perimeters.

The perimeter of any figure is the sum of the lines bounding the figure. In form (6), Fig. 1, if p denote the perimeter, then,

1. What does x + 2x + 4x mean? How may it be more briefly written?

2. What is the coefficient (the multiplier) of x in the answer to problem 1?

3. Write the perimeter, p, of (1), Fig. 1, in two ways. 4. Write the perimeter, p, of (13) in three ways.

5. Write the perimeter, p, of (12) in three ways.

6. Write of the perimeter, p, of (13) in two ways. 7. Write an expression showing that the two answers to 6 are equal.

8. In (12), if x = 80 rods, and y feet are there in the perimeter?

=

40 rods, how many

=

9. In (1) and (2), if x and y each 20 rods and one side of the triangle rests on the square, making a new figure, omitting the common line, what is the perimeter of this new form, in feet? How many sides has the new figure thus formed?

10. Forms (1) and (13) are combined into a single figure. Write p for the new figure in two ways, supposing x the same in both.

§106. Quadrilaterals.

1. Forms (2), (7), (8), (10), (12), and (13), Fig. 1, p. 172, are different kinds of quadrilaterals. What is a quadrilateral?

2. What quadrilaterals have their opposite sides parallel? These figures are parallelograms. Define a parallelogram.

3. What parallelograms have all their sides equal? What is a rhombus?

4. What rhombus has all its angles equal? What is a square?

5. What quadrilaterals have their opposite sides equal but consecutive angles not equal? Define a rhomboid.

6. What parallelograms have their angles all equal? Define a rectangle.

7. What quadrilateral has only one pair of sides parallel? Define a trapezoid.

8. Is a trapezoid a parallelogram? Is it a quadrilateral?