Ex. 1283. What is the locus of the centres of circles which touch a given line at a given point ? Ex. 1284. What is the locus of the centres of circles of radius 1 in., touching a given circle of radius 2 in., and lying inside it? Draw a number of such circles. Ex. 1285. Repeat Ex. 1284 with 1 in. radius for the touching circles, and 3 in. radius for the fixed circle. Ex. 1286. Draw a number of circles of radius 3 in. to touch a circle of radius 2 in. and enclose it. Ex. 1287. Draw a number of circles of radius 4 in. to touch a given circle of radius 2 in. and enclose it. Ex. 1288. What is the locus of centres of circles of given radius passing through a given point ? Ex. 1289. What is the locus of centres of circles (i) passing through two given points, (ii) touching two given lines? Each of the following problems is to be solved by finding the centre of the required circle, (generally by the intersection of loci). Some of the group have been solved already; they are recapitulated below for the sake of completeness. In several cases a numerical instance is given which should be attempted first, the radius of the resulting circle being measured; the student should afterwards deal with the same problems again, drawing the given lines, circles, &c. for himself. Ex. 1290. Draw a circle (or circles) to satisfy the following conditions:(i) To pass through three given points (solved already). (ii) of given radius, to pass through two given points (solved already). (iii) of given radius, to pass through a given point and touch a given line, e.g. take radius 2 in. and a point distant 1 in. from the line. (What is the locus of centres of 2 in. circles passing through given point; touching given line?) When is the general problem impossible? (iv) To touch a given line AB at a given point P, and to pass through a given point Q outside the line. (What is the locus of centres of ing line at P; passing through P and Q? Let PQ=3 cm., <QPA=30°.) touch CONSTRUCTION OF CIRCLES WITH GIVEN CONDITIONS 249 (v) To touch a given circle at a given point P, and to pass through a given point Q not on the circle. In what case is this impossible? (vi) To touch a given line AB at P, and also to touch a given line CD, not parallel to AB. (What is the locus of centres of circles touching AB and CD?) (vii) of given radius, to pass through a given point P and touch a given circle, e.g. let given radius = 4 cm., radius of given circle = 3 cm., distance of P from centre of given circle=5 cm. (Compare (iii).) (viii) of given radius, to touch a given circle at a given point (how many solutions are there?). (ix) To touch three given lines (solved already). (x) Of given radius to touch two sect at an angle of 60°, and radius=1 in. given lines, e.g. let the lines inter(How many solutions are there?) (xi) Of given radius, to touch a given line and a given circle (e.g. given radius = 3 cm., radius of given circle = 5 cm., distance of line from centre of circle 6 cm.). What is the condition that the general problem may be possible? (xii) To touch three equal circles (a) so as to enclose them all, (b) so as to enclose none of them. (Begin by drawing a circle through the three centres.) (xiii) Of given radius, to touch two given circles (e.g. let given radius=2 in., radii of given circles=1 in., 1·5 in., distance between centres =3.5 in.). Ex. 1291. In a semicircle of radius 5 cm. inscribe a circle of radius 2 cm. Measure the parts into which the diameter of the semicircle is divided by the point of contact. Ex. 1292. Draw four circles of radius 2 in., touching a fixed circle of radius 1 in., and also touching a straight line 2 in. distant from the centre of the fixed circle. fig. 233. Ex. 1293. Inscribe a circle in a sector of 60° of a circle whose radius is 4 in. Measure the radius of the inscribed circle. Ex. 1294. Draw three equal circles, each touching the other two; circumscribe a fourth circle round the other three. G. S. II. 17 Ex. 1295. Prove that, if circles are described with centres A, B, C (fig. 228) and radii AY, BZ, CX, the three circles touch. Ex. 1296. A variable circle (centre O) touches externally each of two fixed circles (centres A, B). Prove that the difference of AO, BO remains constant. Ex. 1297. If two circles touch and a line is drawn through the point of contact to meet both circles again, the tangents of its extremities are parallel. (Draw the common tangent at the point of contact.) Ex. 1298. If two circles touch externally at A and are touched at P, Q by a line PQ, then PQ subtends a right angle at A. Also PQ is bisected by the common tangent at A. Ex. 1299. Prove that, in Ex. 1298, the circle on PQ as diameter passes through A and touches the line of centres. Ex. 1300. Two circles intersect at A, B; prove that the line of centres bisects AB (the common chord) at right angles. (See III. 1 Cor.) What kind of symmetry has the above figure? Ex. 1301. Find the distance between the centres of two circles, their radii being 5 and 7 cm. and their common chord 8 cm. (By calculation, and drawing; there are two cases.) SECTION VI. ANGLE PROPERTIES. Reflex angles. Take your dividers and open them slowly. The angle between the legs is first an acute angle, then a right angle, then an obtuse angle. When the dividers are opened out flat, the angle has become two right angles (180°). If the dividers are opened still further the angle of opening is greater than 180° and is called a reflex angle. DEF. A reflex angle is an angle greater than two right angles and less than four right angles. Fig. 235 shows two straight lines OA, OB forming a reflex angle (marked), and also an obtuse angle (unmarked). fig. 234. B fig. 235. A Ex. 1302. Account for the necessity of the phrase "less than four right angles" in the above definition. Ex. 1303. Open a book to form a reflex angle. Ex. 1304. What is the sum of the reflex angle a and the acute angle b in fig. 236? If Lb=36°, what is La? Ex. 1305. What kind of angle is subtended at the centre of a circle by a major arc? Ex. 1306. Draw a quadrilateral having one angle reflex. Prove that the sum of the four angles is 360°. a fig. 236. Ex. 1307. Is it possible for (i) a four-sided figure, (ii) a five-sided figure to have two of its angles reflex? x fig. 237. fig. 238. fig. 239. Ex. 1308. Draw a figure like fig. 237, making the radius of the circle about 2 in. Measure angles x and y. Ex. 1309. Do the same for figs. 238, 239, 240. What relation do you notice between the angle x and the angle y in the four experiments? Ex. 1310. Draw a circle of radius 5 cm.: place in it a chord AB of length 9.5 cm. Mark four points P, Q, R, S in the major arc. Make the necessary joins and measure the angles APB, AQB, ARB, ASB. What relation do you notice between these angles? Can you connect this with the results of Ex. 1308, 1309? A fig. 240. B Ex. 1311. In the figure of Ex. 1310 mark thrce points X, Y, Z in the minor arc; measure the angles AXB, AYB, AZB. Ex. 1312. Draw a circle and a diameter. Mark four points on the circle, at random. Measure the angle subtended by the diameter at each of these points. Ex. 1313. A side BA of an isosceles triangle ABC is produced, through the vertex A, to a point D. Prove that DAC=24ABC=24ACB. |