THEOREM 4. In equal circles (or, in the same circle) (1) if two chords are equal, they cut off equal arcs. (2) Conversely, if two arcs are equal, the chords of the arcs are equal. To prove that Construction Proof arc AGB = arc DHE, and arc ACB = arc DFE. Join PA, PB; QD, QE. In the As APB, DQE Data AP DQ (radii of equal .. the triangles are congruent, .. LAPB = 4 DQE, == .. arc AGB : = arc DHE, Again, whole circumference of ABC chord AB chord DE. Join PA, PB; QD, QE. Ex. 1182. A quadrilateral ABCD is inscribed in a circle, and AB=CD. Prove that AC=BD. Ex. 1183. Prove the converse, in ш. 4, by superposition. Also try to prove the direct theorem by superposition, and point out where such a proof fails. To place in a circle a chord of given length. Adjust the compasses to the given length. With a point A on the circle as centre draw an arc cutting the circle in B. Then AB will be the chord required. Ex. 1184. Place in a circle, end to end, 6 chords each equal to the radius. Ex. 1185. Place in a circle, cnd to end, 12 chords each equal to the radius. Ex. 1186. Draw a circle of radius 5 cm. Place in the circle a number of chords of length 8 cm. Plot the locus of their middle points. Ex. 1187. Construct an isosceles triangle, given that the base is 7 cm. and the radius of the circumscribing circle is 5 cm. (Which will you draw first—the base, or the circle?) Measure the vertical angle. Ex. 1188. Construct a triangle, given BC=3 in., B=30°, radius of circumscribing circle=2 ins. Measure AC and ▲ A. Ex. 1189. In a circle are placed, end to end, equal chords PQ, QR, RS, ST. Prove that PR QS RT. To inscribe a regular hexagon in a circle. A fig. 223. In the circle place a chord AB, equal to the radius. Join A, B to O, the centre. Then AOAB is equilateral, .. LAOB 60°. Place end to end in the circle 6 chords each equal to the radius. Each chord subtends 60° at the centre, .. the total angle subtended by the 6 chords is 360°. In other words, the 6 chords form a closed hexagon inscribed in the circle. Since the chords are equal, the hexagon is regular. (See III. 15.) Ex. 1190. The side of an isosceles triangle of vertical angle 120° is equal to the radius of the circumcircle. Ex. 1191. Find the area of a regular 6-gon inscribed in a circle of radius 2 in. Revise "Regular polygons," Ex. 69-74. Ex. 1192. Find the perimeter and area of a regular 8-gon inscribed in a circle of radius 2 in. CIRCUMFERENCE OF CIRCLE. Take any circular object, such as a penny, a round tin, a garden-roller, a bucket, a running track. Measure the circumference and the diameter; how many times does the circumference contain the diameter? Work out your answer to three significant figures. Methods of measuring the circumference:— (i) Put a small spot of ink on the edge of a penny; roll the penny along a sheet of paper, and measure the distance between the ink spots left on the paper. (ii) Wrap a piece of paper tightly round a cylinder; prick through two thicknesses of the paper; unroll the paper and measure the distance between the pin-holes. (iii) Wrap cotton round a cylinder several times, say 10 times; measure the length of cotton used, and divide by 10. In measuring the diameter, make sure that you are measuring the greatest width. Ex. 1193. Find the value of the quotient circumference for several circular objects of different sizes, and take the average of your answers. Theory shows that the value of this quotient (or ratio) is the same for all circles; it has been worked out to 700 places of decimals and begins thus 3.1415926535....... For the sake of brevity this number is denoted by the Greek letter ; a useful approximation for is 22. The ratios of the perimeters (or circumferences) of regular polygons to the diameters of their circumscribing circles are shown in the following table : : Table showing the perimeters of regular polygons inscribed in a circle of radius 5 cm. It will be noticed that the ratio increases with the number of sides, being always less than π. If the number of sides is very great, the ratio is very nearly equal to . E.g. for a polygon of 384 sides the ratio is 3.14156....... π. Ex. 1194. By how much per cent. does the perimeter of a regular decagon inscribed in a circle differ from the perimeter of the circle? Ex. 1195. Calculate the circumference of a circle whose radius is (i) 7 in., (ii) 14 cm., (iii) 35 miles. (Take π=22.) Ex. 1196. Calculate the circumference of a circle whose diameter is (i) 70 ft., (ii) 21 mm., (iii) 49 miles. |