.. k2=2r. CD=2r(r-OD) = r(2r-20D). 2 But OD = √/AO2 - AD2 = √r2 − (†)2 = ± √4 ̃2 − P2 ; - (1). As a particular case, we may take radius equal to Cor.-Equation (1) expresses k in terms of 7, but it also gives us / in terms of k; for, solving, we have In other words, having given the side of a regular polygon of a certain number of sides, we can find the length of a side of the polygon of half the number. PROP. IX. Having given the side of a regular polygon inscribed in a given circle, to find the side of the regular polygon of the same number of sides circumscribing the E B F circle. Let AB=7, the side of the irscribed polygon; r = radius of the circle. Draw the tangent EF through the middle point C of the arc AB, meeting OA and OB produced in E and F. EF will evidently be the side required, which we shall denote by L. In the similar triangles AOD, EOC, we have EC AD :: OC: OD, As a particular case, we may take radius equal to unity; Cor.-Knowing the side of a regular polygon of n sides inscribed in a circle, we can, by the repeated application of the preceding proposition, find the side, and consequently the perimeter of a regular polygon of 2n, 4n, 8n, 1бn, 32n, &c. sides, inscribed within or described about the given circle. PROP. X.-Given the perimeter of a regular polygon inscribed within a circle, and the perimeter of a similar circumscribed polygon, to find the perimeters of the regular inscribed and circumscribed polygons of double the number of sides. = Let AB side of inscribed polygon; EF, the tangent through the middle cumscribed figure. AC is Join AC. the side of figure inscribed with E double the num K ber of sides. Through A and B draw tangents AG, BH. GH is the side of the figure circumscribed, with double the number of sides. Let p and P denote respectively the perimeters of the inscribed and circumscribed figures; and p' and P' the perimeters respectively of the inscribed and circumscribed figures of double the number of sides. Since OE is the radius of the circle circumscribing the polygon whose perimeter is P, and OC that of the circle circumscribing the similar polygon whose perimeter is p, we have Р OE But since OG bisects the angle EOC, we have (Euclid, VI. 3): OE EG P+ 2p Р = Now, GH is a side of the polygon whose perimeter is P', and is contained in P' the same number of times that EC is contained in P, or Again, the triangles ACD and GKC are similar; But since AD is contained in P the same number of times that AC is contained in p', PROP. XI.-To find the side of a regular decagon inscribed in a circle. In Euclid (IV. 10), it can be shewn that the angle BAD at the centre of the circle is an angle of one-fifth of two right angles, or one-tenth of four; ten such angles can therefore be made round the centre, consequently the line BD is the side of a regular decagon inscribed in the circle. But BD = AC, the greater of the two segments, which is obtained by dividing AB into extreme and mean ratio ; R .. side of regular decagon = (√5−1), Prop. VI. 2 Exercises (2). 1. What must be the radius of a circle in which a chord of 15 inches is 5 inches from the centre? 2. In a circle whose diameter is 20 feet, a chord of 12 feet is inscribed; find the heights of the arcs. 3. The height of an arc is 12 inches, and the chord of half the arc is 36; find the diameter of the circle. 4. The height of an arc is 20 inches, and the diameter of the circle is 60 inches; find the chord of half the arc. 5. In a circle whose diameter is 5 feet 4 inches, the chord of the arc is 20 inches; find the chord of half the arc. 6. In a circle, the chord of half the arc is twice the height; prove that the height is half the radius. 7. The chord of an arc is 20 feet, and the chord of half the arc is 30; find the diameter. 8. In a circle whose radius is R, find (1.) Side of inscribed regular hexagon. 9. In a circle whose radius is R, shew that the side of the regular inscribed octagon 10. In a circle whose radius is R, shew that the side of the regular inscribed dodecagon = R√√2 - √3. II. Compare the perimeters of the inscribed equilateral triangle, the square and the octagon. 12. Find the radius of a circle in which two parallel chords of 6 and 8 inches are 1 inch apart. 13. What is the length of the tangent drawn from a point 10 inches from the centre of a circle whose radius is 8 inches? 14. How far from the centre of a given circle must a point be taken, through which if a tangent be drawn, the tangent will be equal to (1.) The radius of the circle? (2.) The diameter of the circle ? 15. The distance between the centres of two circles, whose radii are R and r, is D; find the lengths of their common tangents. What do these lengths become when D=R+r? 16. Find the side of a regular decagon inscribed in a circle whose radius is R. 17. If d= the diagonal of a regular pentagon inscribed in a circle whose radius is R, prove that R d= 10+2 √5. |