A line segment drawn from the center to a point on the circle is a radius (Fig. 187). A line segment drawn through the center and terminated by

arc

FIG. 187

the circle is a diameter (Fig. 187).

A portion of a circle, as AB (Fig. 187), is an arc.

101. Notation for circles and arcs. The symbol for circle is o. The symbol for circles is . Thus, OA means: a circle whose center is A.

The symbol for arc is. "Arc AB" is written AB.

THE USE OF THE CIRCLE IN DESIGNS

102. How to draw a circle with the compass. A circle is more easily drawn with the compass (Fig. 188) than with a piece of cardboard, as shown in $100. When drawing a circle, hold the compass with one hand, between the thumb and forefinger, and in turning the compass press down lightly on the sharp point.

The exercises found on the following pages give practice in drawing circles:

FIG. 188

EXERCISES

1. Draw several circles on paper. With the blackboard compass draw circles on the blackboard.

2. Name several objects whose boundary lines are circles.

3. On a sheet of paper mark a fixed point A. Locate all the points on the paper which are 5 cm. from A. What seems to be the location of these points?

4. Draw a circle whose radius is 3.5 centimeters.

5. Draw a segment AB, 6 cm. long.

Using A as center and
Using B as center and

a radius equal to 2 cm., draw a circle.
a radius equal to 4 cm., draw a second circle.

6. Draw a segment AB,

8 cm. long. Then construct the design shown in Fig. 189.

7. Draw the design

A

4cm. C 4cm.

B

shown in Fig. 190, making AB equal to 8 centi

meters.

8. Draw a spiral (Fig. 191).

9. Using ruler and protractor, draw a square whose side is 4 centimeters. Construct the designs shown in Figs. 192 and 193.

Fig. 194

11. Draw the design (Fig. 195) using your own ideas as to shading.

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FIG. 195

12. Designs like those shown in Figs. 196 to 204 were made by high-school pupils. Make a copy of one of them. Try to make an original design as good as these, or better.

13. Draw an equilateral triangle.

Construction: Draw a segment AB (Fig. 205). With A as center and radius AB draw an arc, as shown at C.

With B as center and radius AB draw a second arc intersecting the first at C.

Draw AC and BC.

FIG. 204

Triangle ABC is the required equilateral triangle.

FIG. 205

This construction is important because it is used in many designs,

14. Fig. 206 is a picture of a Gothic window. The design is based on the Gothic arch (Fig. 207), as ACB and AED. The construction is the same as that of Fig. 205. The arcs AC and BC are

drawn with B and A as centers and a radius equal to AB. Make a drawing of the arch.

15. The range of the guns of a fort is 10 miles. By a scale drawing show the ground covered by them.

16. Two forts are 18 miles apart. The range of their guns is 12 miles. Make a scale drawing showing the ground covered by the guns of the two forts.

17. A cow is tied by a rope 24 feet long to a stake placed at the mid-point of the longer side of a shed 12 feet by 10 feet in dimensions. Make a drawing showing the ground on which the cow may graze.

USES OF THE CIRCLE IN LATITUDE, LONGITUDE,

AND TIME

103. Equal circles. Draw two circles using the same radius, one on ordinary notebook paper and the other on thin tracing paper. Place the tracing-paper circle over the other circle, and show that the two circles can be made to fit exactly (coincide). If two circles can be made to coincide they are equal circles.

The experiment above illustrates the fact that two circles are equal if they have equal radii.

Two equal circles may be considered as the same circle in two different positions. Hence, radii of equal circles are equal.

104. Central angles. Draw two equal B and B1 (Figs. 208 and 209), one on notebook paper and the other on thin tracing paper. Draw angle B having the vertex at the center B, and draw angle B1 equal to angle B.