Exercises. 1. Describe a circle; draw in it any chord; join the centre to the extremities of the chord; and drop a perpendicular from centre on chord. What is the relation between the angles the radii make with the chord? What between the angles the radii make with the perpendicular? What between the segments of the chord made by the foot of the perpendicular? 2. With the bevel or protractor construct two equal angles at the centre of a circle, and draw the chords which subtend these angles. What is the relation between these chords? Apply test. Give reasons. 3. Describe a circle, and with dividers and ruler place two equal chords in it. Join the ends of the chords to the centre. What is the relation between the angles these equal chords subtend at the centre? Apply test. Give reasons. 4. As in the previous question, in a circle place two equal chords, and from the centre drop perpendiculars on them. What is the relation between these perpendiculars? Apply test. Give reasons. 5. Describe a circle of radius 3 inches, from the centre draw two equal lines of length 2 inches, and through the extremity of each draw a line at right angles to it, so obtaining two chords at equal distances from the centre. What is the relation between the lengths of these chords? Apply test. Give reasons. 6. The sides of a triangle are 2, 3 and 3 inches. circle passing through the angular points. 7. The sides of a triangle are 2, 3 and 4 inches. passing through the angular points. Describe a Describe a circle 8. The sides of a triangle are 3, 4 and 5 inches. Describe a circle passing through the angular points. 9. Two chords of a circle with one end of each common, are of lengths 2 and 3 inches, and make an angle of 60° with each other. Describe the circle. 10. Two chords of a circle make angles of 50° and 60° with a third chord whose length is 23 inches, and are inclined towards one another. Describe the circle 11. The sides of a rectangle are 40 and 60 millimetres. circle passing through all the angular points. Describe a 12. Describe a parallelogram ABCD, not being a rectangle. Can a circle be described passing through its angular points? (Every circle through A and B has its centre on the line which bisects AB at right angles.) 13. The diameter of a circle is 30 inches, and a chord is 24 inches. How far is the chord from the centre? 14. The radius of a circle is 3 inches. What is the length of a chord whose distance from the centre is 14 inches? 15. The equal sides AB, AC, of an isosceles triangle ABC, are 50 millimetres, and they contain an angle of 45°. A circle with centre A, and radius 70 millimetres, cuts BC produced in D and E. What is the relation between the lengths of DB and CE? Apply test. reasons. Give 16. Describe a circle; draw a diameter, producing it; and from a point A in the produced diameter draw two lines on opposite sides of it, making equal angles with it. What do you observe as to the lengths of the segments of these lines between A and the points of section by the circle? What as to the parts within the circle? Apply tests. Give reasons. 17. The same question as the preceding, but with A within the circle. 18. Construct two intersecting circles, join their centres, and through either of the points of intersection, draw a line parallel to the line joining centres, and terminated by the circumferences. What relation in length between the second line drawn and the line joining the centres? Apply test. Give reasons. 19. AB, CD are two parallel chords in a circle. What relation exists between the lengths of the chords AC, BD? Apply test. reasons. Give 20. In the previous question prove angle ABD=angle BAC; also angle ACD=BDC; also chord AD chord BC. CHAPTER XI. Tangents to Circles, and Circles Touching One Another. 1. To draw the tangent at any point A on the circumference of a circle, draw the diameter through A, and draw at A the perpendicular to this diameter. The perpendicular is a tangent to the circle (Ch. X.,. 2). Evidently the tangents at opposite ends of a diameter are parallel to one another. Draw Construct a circle of radius 55 millimetres. radii at intervals of 30°, and draw the tangents at the ends of these radii, producing each both ways until it meets the adjacent tangents. Construct a circle of radius 49 millimetres. Draw radii at intervals of 45°, and draw tangents at the ends of these radii, producing each both ways until it meets the adjacent tangents. Construct a circle of radius 15% in. Draw radii at intervals of 72°, and draw tangents at the ends of these radii, producing each both ways until it meets the adjacent tangents. In each of the three preceding constructions, the resulting figure about the circle should have equal sides and equal angles. The equality of the sides (measured with the dividers) and the equality of the angles (measured with the bevel) may be regarded as a test of the accuracy of the construction. Any two diameters in a circle are drawn, inclined at an angle of, say, 30° to each other, and tangents at the ends of these diameters are constructed. What quadrilateral figure about the circle do the tangents form? Measure its sides. 2. From a point without a circle, evidently two tangents can be drawn to the circle. To draw those from A to the circle FBG: E Join AC, cutting the circle in B. Describe a second circle DAE, with centre C and radius CA. Draw DBE perpendicular to CB. Join CD and CE, cutting the small circle in F and G. Then AF and AG are the tangents from A. For, the triangles ACF and DCB are equal. But the angle CBD is a right angle; therefore the angle CFA is a right angle, and AF is a tangent to the circle (Ch. X., 2). In the same way we may prove that AG is a tangent. Symmetry suggests that the tangents AF, AG are equal in length, and that they make equal angles with AC. The truth of this may be tested by measurement.. It may also be proved as follows: Because CDE is an isosceles triangle, and the angles at B right angles, therefore the triangles CDB, CEB are equal in all respects. But the triangle CAF is equal in all respects to CDB; and the triangle CAG is equal in all respects to the triangle CEB. Therefore the triangles CAF and CAG are equal in all respects. Hence AF, AG are equal, and the angles at A are equal. In practice, an easy way to draw a tangent from any point A, outside the circle, is as follows: Place the setsquare so that one of its sides passes through A and the other through C, the centre of the circle. Then so adjust the instrument that B the right angle rests on the circumference at, say, B. AB, a tangent through A, may then be drawn. Construct a circle of radius 1 in., and draw any line through its centre. From points on this line at distances from the centre 2, 2, 3 in., draw tangents to the circle. 3. Let a circle be described with centre A, and the tangent at any point C be drawn; and let, with centre B, on AC, and radius BC, another circle be drawn. Then both circles have CD for tangent. Both touching the same line at the same point, they are A B D said to touch one another, in this case internally. |