Repeat the same examination in the case of different circles, drawing the chord at various inclinations to the tangent. As a result of these observations we are led to the conclusion that if from the point of contact of any tangerit to a circle, a chord be drawn cutting the circle, the angles the chord makes with the tangent are equal to the angles in the alternate segments of the circle. We may establish the same result in the following way: Let AG be the diameter through A. The angles GED, GAD, GFD, are equal to one another because they stand on the same arc GD. Also the angles CAG, BAG, AEG, AFG are right angles. Hence B BAG - DAG= LAEG - DEG, or ZBAD= AED, in alternate segment. Again, ZCAG + ZDAG= _ AFG + LGFD, or / CAD= LAFD, in alternate segment. It will be noticed that, as AD revolves to the right about A, the angles BAD, AED, have just as much taken from them as CAD, AFD have added to them, the points E and F being supposed to remain stationary. Placing the centre of the protractor on the circumference of a circle, and marking the initial line of protractor as a chord, we may place in the circle an angle of any required magnitude, i.e., we may cut off from the circle a segment containing an angle of any size. Exercises. a a a 1. Describe a circle of radius 13 in., and in it place an angle of 60°. In it also describe a triangle of vertical angle 60° and altitude 2 in. 2. Describe a circle of radius 35 millimetres. From it cut off a segment containing an angle of 50°, and describe in it a triangle with angles 50°, 30° and 100°. 3. Describe a circle of radius 40 millimetres, and in it describe a triangle with angles 50°, 55° and 75°. 4. Describe a circle of radius 2 in. Draw a chord AB, cutting off a segment containing an angle of 120°, and a chord BC, cutting off a segment containing an angle of 100°. What is the angle contained in the segment cut off by CA? Apply test. Give reason. 5. Describe a circle of radius 50 millimetres, and in it draw a chord cutting off a segment containing an angle of 55°. What angle is contained in the segment which forms the rest of the circle? Apply test. Give reason. 6. Describe a circle of radius 19 in. Draw in it a chord AB, dividing the circle into two segments, ACB, ADB, containing angles of 70° and 110° respectively. Construct in the circle an angle CAD of 50°. What is the angle CBD? Mark on the quadrilateral ACBD the size of each angle. 7. Describe a circle of radius 40 millimetres, and in it construct a quadrilateral with angles 55°, 75°, 125°, 105°. 8. Describe a circle of radius 45 millimetres, and in it draw a number of chords, AB, CD, EF, . . all cutting off angles of 60°. Are the chords all of the same length ? Apply test. Give reasons. 9. Describe a circle of radius 1} in., and in it construct a triangle with angles 30°, 70°, 80°. Does the size of the triangle vary according as it happens to be placed in the circle ? Give reasons. 10. Describe a circle of radius 2 in., and in it construct a quadrilateral with angles 45°, 120°, 135°, 60°. Show that the size and shape of the quadrilateral can be made to vary. What lines belonging to the quadrilateral remain constant ? 11. ABCD is a quadrilateral in a circle, and the side AB is produced to E. To what angle of the quadrilateral is the exterior angle CBE equal ? Apply test. Give reasons. a · 12. AB is a line of length 24 in. If on it a segment of a circle is to be constructed containing an angle of 60°, what angle will AB subtend at the centre C ? What are the angles of the triangle CAB? Find C by construction, and then describe the circle. 13. AB is a line of length 60 millimetres. Following the method suggested in the previous question, construct on it a segment of a circle containing an angle of 70°. Test the accuracy of your construction by measuring an angle in the segment. 14. AB is a line of length 24 in. ; to construct on it a segment of a circle containing an angle of 70°: Make BAC=90°, ABC=90° – 70°= 30°. Then ACB=70°. Bisect BC at 0, and with O as centre and OA, OB, or OC as radius, describe a circle. The segment ACB contains an angle of 70°, and it stands on AB. 15. Construct a triangle with sides 60, 75 and 85 millimetres. On these sides, and within the triangle, construct segments containing angles of 120°. Should these segments all pass through the same point within the triangle ? 16. AB, CD are two chords, perpendicular to each other, in a circle whose centre is O. Of what angles are the angles AOC, · BOD double? What, therefore, is their sum ? 17. AB, CD are two chords of a circle, intersecting in E. Show that the triangles AEC, DEB are equiangular. 18. ABCD is a quadrilateral in a circle, and the sides AB, CD, produced, meet in E. Show that the triangles EBC, EDA are equiangular. 19. AB, AC are tangents to a circle whose centre is O. Show that BOC= 180- A ; also that the angle in the segment BC, between the tangents, contains an angle 90° +3 A. 20. AD, BE are drawn perpendicular to the opposite sides of the triangle ABC. Show that a circle can be described about AEDB, and describe it. How are the angles ABC, DEC related ? Apply test. Give reasons. CHAPTER XIII. Relation Between Segments of Intersecting Chords. B F K 1. AEB and CED are any two chords in a circle, intersecting at E. In the second figure CEB and AED are any two lines drawn E perpendicular to each other, and and on these we lay off the following distances with the dividers : H B ED=ED 66 Complete the rectangles CEDF and AEBG, and let FD and GB meet in H. Then produce the lines FC, HE and GA, and note how nearly they come to passing through the same point (at K). Go over the measurements and construction with extreme care, getting rid of all inaccuracies. Do these lines (FC, HE, GA) all pass through the same point? If they do, how do the areas CEDF, AEBG compare in size (Ch. VIII., 6), and therefore the rectangles AE.EB, CE.ED, contained by the segments of the chords ? Measure the number of millimetres in each of the lines AE, EB, CE, ED in the circle, and examine whether the product of AE and EB is approximately equal to the product of CE and ED. Describe other circles, draw two chords in each, and repeat in the case of each circle the construction of the second figure. Repeat also the measurements and multiplications. The result of our observations may be stated as follows: If two chords of a circle cut one another within the circle, the rectangle contained by the segments of the one is equal to the rectangle contained by the segments of the other. 66 I 66 2. Draw accurately the tangent EC ; draw also the secant EAB. In the second figure CEA, BEC are any two lines drawn at right angles to each other, and on these we lay off the following distances with the dividers : EA=EA of circle F к care, getting rid of all inaccuracies. Do these lines (FC, HE, GB) all pass through the same point? If they do, how do the areas EBGA, ECFC compare in size (Ch. VIII., 6), and therefore the rectangle EA.EB and square on EC (see figure of circle) ? A |