The square, triangle, rectangle and parallelogram are the basis of many decorative designs.

1. Draw a square 2 in. on a side. Divide each side into four equal parts. Join the corresponding points in the opposite sides by dotted lines. Draw heavy lines, as in the figure. Erase the dotted lines. Shade in two ways or color in two colors.

2. Draw an arc of a circle and connect its ends by a straight line.

Chord. The straight line connecting the two ends of any arc is called the chord of that arc. The chord is said to subtend the arc.

ARC

CHORD

CHORD

ARC

3. What is the longest chord in any given circle?

4. Draw a circle. By means

of the protractor draw diameters making angles of 60°. Connect the ends of these diameters

alternately by chords, as shown in the figure. Join the center with the intersections of the chords. Erase all lines except those corresponding to the heavy lines of the figure. A six-pointed star results. Shade in two ways or color in two colors.

5. Draw a square; bisect the sides; connect these points in order about the square. Color or shade the triangular parts.

✡

6. Fold or weave two narrow strips of paper of equal length in the form shown at the left; of what two figures is it composed? If the side of one of these is 2 in. long, how many inches of paper are needed

to make the design?

1. What is a six-sided plane figure called?

The regular hexagon occurs frequently in nature, and is much used in architecture and designs. It is constructed by drawing a circle and drawing successive chords equal to

Fig. 1.

the radius. (See Figure 1.)

2. Construct a regular hexagon using dotted lines for the sides.

3. From the center of the figure draw full lines to the vertices.

Along each such line draw

short parallel lines as shown in Figure 2. The resulting figure is a common pattern among snow crystals.

4. Draw a regular hex

agon in dotted lines.

Fig. 2.

By

Fig. 3.

using each vertex as a center, describe the small arcs shown in Figure 3. Complete the figure and erase the dotted lines.

5. Show how to draw the designs of these figures:

6. Explain from the figures how to construct a design for each of these moldings:

Oral.

PROBLEMS-REVIEW

1. What is an angle? Illustrate your answer.

131

2. Explain how to construct a triangle whose sides shall be equal in length to three given lines; also how to construct an equilateral triangle having its sides equal to a given line.

3. What is an isosceles triangle? If the angle at the vertex is 80°, what is the size of each angle at the base?

4. What is the sum of the angles of a triangle? When the three angles of a triangle are equal, how many degrees in each?

5. If the three sides of one triangle are respectively equal to the three sides of another triangle, how are the triangles related? How if two sides and the included angle of one are respectively equal to two sides and

the included angle of the other?

6. Two points are 8 in. apart; how may a point be found that is 10 in. from one of these points and 5 in. from the other?

Written.

7. The figure shows the plan of a basement wall. How many cubic yards of stone will be required to build the wall 8 ft. high and 18 in. thick?

LAGOON

MI.

PARK LANE

MI.

BASIN

UXBRIDGE

ROAD

1-2 FT:

---20 FT

FT.

-50 FT.

8. The main hall of the Providence railway station is 180 ft. long and 44 ft. wide; what is the area of the floor?

9. The train shed of the same station is 588 ft. long and 130 ft. wide; what area does it cover?

10. The figure shows the general shape and size of Kensington Gardens,

Hyde Park, London; regarding it as a trapezoid, find its area.

SOLUTION OF PROBLEMS

ANALYSIS

How to Solve Problems. The four steps in the solution of a problem have already been explained and applied (Book II, pages 92, 93), namely:

1. Read the problem.

3. Make the calculation.

2. Plan the solution.
4. Test the work.

The analysis of a problem is the plan of the solution. It may be expressed orally or written in steps.

EXAMPLE. A dealer buys Welsbach lamps at $9.00 a dozen; for how much must he sell them to gain 50 cents each?

Oral Analysis. 2 of $9.00, or $.75, is the cost of 1 lamp. $.75, the cost, plus $.50, the gain, is $1.25, the selling price of the lamp.

Solve these problems, writing the analysis in step form and the calculation; also give the oral analysis:

1. A man bought a hat for $2.50 and 6 collars at 15¢ each; how much did he pay for these articles?

2. 31⁄2 lb. of coffee cost $1.12; what was the cost 1 lb.? 3. A dealer paid $30 for a dozen pairs of shoes; he sold them at a gain of $.75 a pair; find the selling price a pair?

4. Taking 500 bunches of lath as a carload, how many carloads would 3,500,000 bunches make?

5. How many times would the water in a cistern containing 5,180 gal. fill a bucket that holds 21 gal.?

NOTE.-The oral exercises provide opportunity throughout the book for drill in oral analysis.

UNITARY ANALYSIS

133

Important Methods. There is no single plan that will apply to the solution of all problems of arithmetic. But there are two methods widely used, unitary analysis and the equation. We have already used both methods many times; they will be reviewed here.

Unitary Analysis. Unitary analysis may best be explained by a few examples:

(1) What is the cost of 45 bottles of ink at $3.00 per hundred?

ANALYSIS.

1. 100 bottles of ink cost $3.00.

2. Therefore, 1 bottle costs of $3.00, or $.03.
3. Therefore, 45 bottles cost 45 x $.03, or $1.35.

(2) If 14 bu. of peaches make 270 qt. of fruit for canning, how many qt. will 63 bu. make?

ANALYSIS.

1. 14 bu. make 270 qt.

2. Therefore, 1 bu. makes of 270 qt., or 194 qt.

3. Therefore, 63 bu. make 63 × 194 qt., or 1215 qt.

It is usually better first to indicate all the work, then cancel as much as possible, and multiply last of all.

The solution of example (2) would appear thus:

I. If 25 yd. of cloth make 15 vests, how many yards are needed to make 36 vests?

2. If a steamer goes 9 mi. in 48 minutes, how long will it take to go 51 mi. at the same rate?

3. If the interest on $330 is $22, how much is the interest on $780 for the same time and rate?

4. If 16 men can dig 980 yd. of sewer trench in 8 days, indicate the amount they can dig in 1 day; the amount 1 man can dig in 1 day; in 20 days; the amount that 28 men can dig in 20 days. Find the last named amount.