case (Fig. 11) the angle ABP between two sides of the quadrilateral ABPQ in a circle is equal to the external angle AQO between the other two sides, and in the same way LBPQLQAO. So that the As OBP and OQA are Ex. 10. From measurements of each of your figures calculate Ex. 11. Prove the proposition for two positions of OP, neither position passing through C. 18. The length of the radius being r cm. and the distance of O from the centre being s cm., express OA, OB, and OAX OB in terms of r and s. -Lengths being in cm. and areas in sq. cm., we have in Fig. 10, OA X OB = (r-s) × (r+s) = rr—ss, or r2-s2. In Fig. 11 OA = s―r, OB=s+r, and OA × OB = (s—r) (s+r) = s2 — r2. Express OPX OQ in terms of s and r. Ex. 12. Measure r and s on each of your figures, calculate the difference between 2 and s2, and compare it with the values you found for OP× OQ for various positions of CP. Ex. 13. Draw a rectangle with sides r+s centimetres and r centimetres, and show how to cut it up and rearrange it as a square r centimetres in the side, with a little square 8 centimetres in the side cut away. 19. Consider the case of O inside the circle. On the diagram that holds your graph of OP as a function of the angle turned through by CP, draw a graph giving OQ as a function of the angle turned through by CP. For what positions is OQ greater than OP, and for what positions is it less?-Draw the chord MON 1 to AOB. When P lies on the greater arc MBN, OQ is less than OP; when P lies on the less arc MAN, OQ is greater than OP; and when P = Express OMX ON in and s = 7, find the We could find the length of OM by drawing the figure and measuring. Or, CO being the 1 from the centre of the circle on the chord MN we know that OM is ON. So OM× OM, or OM2, is equal to 2-s2, that is, 81-49, or 32. From the graph of x2 (chap. III, art. 17), we find that approximately 32, so that OM = 56; all distances being in centi = 5.62 metres. Suppose in general OM to contain k centimetres. Write down the relation between k, r, s. -It is k2 2-s2, or r2 = k2+s2, or $2 s2 = 12-k2. If a triangle is made = = with these lengths k, r, s as sides, what do you know of its shape? Draw it for the case r = 9, 8 = 7, k 5.6.—The triangle is right angled, for it is the triangle COM of Fig. 10. Ex. 14. Prove that if AOCB is a diameter of a circle and OM drawn to AOCB meets the circumference in M, then OM × OM · OA × OB. (Do not simply quote the main proposition but prove this case by considering similar triangles.) = Ex. 15. Show how to draw a square equal in area to a given rectangle. Ex. 16. Show how to draw a square equal in area to a given triangle. Ex. 17. Draw a line representing 10 on any convenient scale. Show how to draw another line to represent on the same scale a number that multiplied by itself is equal to 10. 20. Consider the case of O outside the circle. On the same diagram that contains your graph of OP, give a graph showing OQ as a function of the angle CP has turned through. Distinguish when OP is greater than, less than, and equal to OQ. Q and P travel round the circle in opposite directions. There are two points G and H at which they pass one another, and these two points divide the circle into two arcs. When P is on the greater arc, OP > OQ; when P is on the less arc, OP < OQ; and when P is at G or H, OP : What is a line such as OG or OH that meets a circle in only one point? You have expressed OPX 0Q in terms of r and s. What does = OQ. this relation become at the points G and H where P and Q pass one another ?—It is OG × OG = s2—r2. 21. How would you draw a tangent to a circle, centre C, from a point 0 outside it (Fig. 12) ?—We could lay a ruler against the circle and through 0 and so draw the tangent. That method gives the tangent very well, but does not give the point of contact very accurately.— G being the point of contact we know that LOGC is a right angle, so that G must lie on a circle on OC as diameter. We draw this circle and so find the two points of contact of tangents from 0. By considering similar triangles, find a relation between OG, OA, and OB.-Join GA and GB. The angle OGA between the tangent GO and the chord GA is equal to the angle GBA in the alternate segment. So the two triangles OAG and OGB have LOGA LOBG, and the angle at O is an angle of each triangle. The triangles are therefore ОА OG similar, and = or OG2 = OA × OB. 22. We saw earlier that the sign of multiplication x could sometimes be omitted without loss of clearness. In the present case confusion would result from its omission, but a dot may be used without loss of clearness. The above relation is then written OG2 = OA.OB. = OQ ов in Figs. 10 and 11, is called a proportion, and the four lengths OA, OQ, OP, OB are said to be in proportion. When the proportion contains the same length twice, as a numerator in one ratio and as a denominator in the other, this length is called a mean proportional between the other two. Thus OG in Fig. 12 is a mean proportional between OA and OB. = = 23. Suppose the length of the tangent OG to be tcm. and give a relation between t, r, s.—OA s-r, OB = s +r, t2 OA.OB = s2 —p2. It may also be put in the forms s2 r2+t2, and 2 s2-t2. What sort of triangle do the three lengths r, s, t form ?—They form the right-angled triangle OGC. = = 24. Sum up your conclusions with regard to the rectangle properties of a circle.-If a line is drawn through a point O and cuts a circle in P and Q, the area of the rectangle contained by OP and OQ is independent of the direction in which the line is drawn through 0. If the circle has a radius of r cm. and O is scm. from the centre; then the area of the rectangle is r2-s2 sq. cm. when O is inside the circle, and s2-r2 when O is outside. Also when O is outside, the area of the rectangle is equal to the square on the tangent from 0 to the circle. EXERCISES. 18. Draw a circle with a triangle ABC inscribed in it. Draw the creases that appear when (i) B is folded on to C; (ii) AB is folded on to AC. Do the creases meet on the circle? Test by a geometrical proof. 19. A regular hexagon is inscribed in a circle. Express, in degrees, the value of the angle at the centre subtended by each side. A paper circle is creased so as to show a diameter AB, and the circle is then folded so that the points A and B are brought together to the centre, two other creases being thus obtained at right angles to AB. Prove that the six points now marked on the circum ference by the three creases are angular points of a regular hexagon. You may show this, if you choose, by construction and measurement, but a general proof should also be given. 20. Three straight pieces of railway track meet at three junctions A, B, C, thus forming a triangle. Where would you place a gun so as to cover the three tracks? If your gun won't carry far enough to cover the three junctions at once, where will you place it to command some part of each track? 21. A gasometer is in the form of a cylinder with a rounded top, and the following dimensions are given: diameter = 28 feet, height at edges 14 feet, height in the middle 16 feet. Draw a section of the gasometer on the scale of 1 inch to 10 feet, and calculate the number of cubic feet of gas that the gasometer will hold. Take the volume of the portion at the top (above 14 feet) as half the volume of a cylinder of the same base and height, and the area of a circle as 3·14 times the square on its radius. 22. There are shoals off a coast on which stand two prominent objects P and Q. A chart of the coast shows two lines from P and Q meeting at an angle a and marked danger angle'. What single observation might be taken on a ship when within possible range of danger to test if it was outside the danger zone ? 23. Take three points L, M, N, and suppose them to represent the positions of three lighthouses along a stretch of coast LMN. A ship O is off the coast, and the angles LOM and MON are found to be 80° and 75°. Construct the position of the ship, give no proof, but point out why there is only one solution. Prove the truth of a property of the circle that you employ. 24. A barometer tube, whose internal diameter is 2 centimetres, is 120 centimetres long and is filled with mercury. Find the weight of the mercury, being given that a cubic centimetre of mercury weighs 13.6 grams, and that the area of a circle is 0.785 x (diameter)2. Also find the cost of the mercury at 6s. 3d. a kilogram. 25. State how you would proceed if you were required to draw about a given circle a quadrilateral, which is to be such that a circle may be drawn about it. Briefly justify your method. To what extent is the shape of the quadrilateral at your disposal? 26. A canal-lock is to be made 100 feet long and 17 feet wide, and to let a boat pass through up-stream the water in the lock must rise 8 feet. The sluice to admit the water is to be designed |