with the protractor. The segment containing an angle of 30° may also be obtained by constructing at the centre an angle of 60°.

In a circle whose radius is 45 millimetres, construct a triangle whose angles are 75°, 45° and 60°.

In a circle whose radius is 1 in., construct a triangle whose angles are 65°, 75° and 40°.

2. To construct a triangle whose sides shall be tangents to a given circle, and whose angles shall be of given magnitude, say, 75°, 45° and 60°.

We can scarcely here proceed as in the previous case, adjusting the legs of the bevel to, say, the angle 75°, and placing them across the circle so as to be tangents to it. To assume that we can construct the tangent to a circle by laying the ruler against it and so drawing a line, is equivalent to assuming that we can lay off a right angle, using only the judgment of the eye.

It will be well to proceed thus: Find the angles which are the supplements of 75°, 45° and 60°, i.e., 105, 135° and 120°. Draw

any radius OA, and make the angle AOB of 105°, and the angle AOC of 135°.

The re

maining angle BOC must be of

120°, since 105° +135° + 120°

360°. Draw lines (tangents)

at A, B and C at right angles E to the radii.

·20°

105°

60

D

B

A

75°

Since the angles of a quadrilateral make up four right angles, and the angles at A and C are right angles, therefore AOC+AEC=180°. But AOC is 135°.

Therefore. AEC is 45°, if AOC has been accurately constructed, and the tangents at A and C correctly drawn. Similarly the angles at D and F are 60° and 75° respectively.

The triangle DEF is said to have been described about the circle.

About a circle whose radius is 20 millimetres, construct a triangle whose angles are 70°, 80° and 30°.

About a circle whose radius is 35 millimetres, construct a triangle whose angles are 90°, 30°, and 60°. About a circle whose radius is 14 in., construct an equilateral triangle.

About a circle whose radius is 1 in., construct an isosceles triangle whose vertical angle is 30°.

3. In a circle we readily place a chord of any required length. For, take the length on the ruler with the points of the dividers, and place the points of the dividers on the circumference of the circle. The ends of the chord, A, B are thus marked, and the chord can be drawn. We can without difficulty

draw the chord in a required position, for example, parallel to a given line, KL Draw OC perpendicular to KL, and mark off CD, CE each equal to half the length of the chord. Then draw DB, EA, parallel to CO.

B

The chord AB is equal to ED, and therefore is of the required length, and it is parallel to KL.

We may draw EA alone perpendicular to KL, and then draw AB parallel to KL, thus not using the point D or line DB.

Of course the chord can never be greater than the diameter of the circle in which it is to be placed.

In a circle whose radius is 55 millimetres, draw chords, with one end at the same point, of lengths 20, 25, 30, 35, 40, 45, 50, 55 and 110 millimetres.

In a circle of radius 1 inch, place ten chords of length inch, such that each ends at the point where the next begins.

In a circle of radius 30 millimetres, place six chords each of length 30 millimetres, such that each ends where the next begins.

In a circle place a chord of given length so that it may be perpendicular to a given line.

Exercises.

1. In a circle of radius 45 millimetres, place an angle of 35°; also an angle of 145°.

2. In the circle of the previous question place these same angles so that the chord or chords on which they stand may be parallel to a line that makes 45° with the edge of your paper.

3. In a circle of radius 2 in., place an angle of 50°, so that the chord on which it stands may be perpendicular to a line that makes an angle of 60° with the edge of your paper.

4. In a circle of radius 1 in., place in succession four chords, AB,

5. In a circle of radius 1 in., construct an equilateral triangle.

6. In a circle of radius 2 in., construct an isosceles triangle, the angle at the vertex being 55°. (Construct at centre an angle of 110°. The symmetry of the circle suggests the rest of the construction.)

7. In a circle of diameter 3 in., construct an equilateral triangle, such that its base shall be parallel to the top or bottom of your paper. (Draw a line through centre perpendicular to top or bottom of paper, and at centre construct, on each side of this line, angles of 60°. Etc.)

8. Construct a triangle with angles of 55°, 65°, and 60°, and in a circle whose radius is 12 in. construct a triangle equiangular to this, its sides being also parallel to the sides of this triangle.

9. Describe a circle of radius 48 millimetres, and draw a line making an angle of 45° with the edge of your paper. Construct a triangle with angles 48°, 75°, and 57°, so that the side opposite 48° may be parallel to the line.

10. Describe a circle of radius 40 millimetres, and draw a line making an angle of 60° with the side of your paper. Draw a tangent to the circle parallel to this line. (From centre drop a perpendicular on the line. This gives point through which tangent is to be drawn.)

11. Describe a circle of radius 35 millimetres. Draw a line making an angle of 75° with the top or bottom of your paper, and draw a tangent to the circle perpendicular to this line. (Draw perpendicular to line, and then tangent parallel to this perpendicular.)

12. About a circle of radius 1 in. describe an equilateral triangle.

13. Describe a circle of radius 35 millimetres, and about it describe an equilateral triangle so that two of the sides may make angles of 60° with the side of your paper, the third side being parallel.

14. Describe a circle of radius 25 millimetres, and about it describe an isosceles triangle whose vertical angle is 40°, the base of the triangle being parallel to the top or bottom of your paper.

15. About a circle of radius 14 in. describe a triangle whose angles are 30°, 70° and 80°.

16. Draw any three intersecting lines. Describe a circle of radius 13 in., and about it describe a triangle whose sides are parallel to the lines. Test the accuracy of your construction by comparing the angles of the two triangles.

17. When a triangle ABC is inscribed in a circle, what are the magnitudes of the angles which the sides subtend at the centre compared with the magnitudes of the angles of the triangle?

18. Describe a circle of radius 1 in. In and about it describe two triangles with angles 50°, 60° and 70°, so that corresponding sides are parallel to each other.

19. An equilateral triangle is inscribed in a circle, and another is described about the circle. What relation exists between the lengths of the sides?

20. Describe a circle of radius 32 millimetres, and draw two tangents to it, such that the angle between them is 25°.

21. Describe a circle of radius 1 in., and from the same point draw two tangents to the circle, each of length 3 in.

22. Describe two circles of radii 1 in. and 2 in. In them describe triangles with angles of 45°, 65° and 70°. Compare the lengths of corresponding sides of the two triangles.