Draw GH parallel to BC, then since the sides about the equal angles in equiangular or similar triangles are proportional, we have in the similar triangles CEF and HDG, FE: EC.:: GD : DH=\, which added to AD will give AH=*. Also in the sim. A's AGH, ABC, AH : AC :: GD: BI=15.36, and this multiplied by half the base will give the area of ABC=245.76. Again in each of the right angled triangles ADG and CEF we have the two legs to find the hypothenuses AG= 5 and CF=10. Now by sim. A's ADG, AIB, GD : GA :: BI : BA=19.2; and FE : FC :: BI : BC. 2. If from the right angled triangle ABC, whose base is 12, and perpendicular 16 feet, be cut off, by a line DE parallel to the perpendicular, a triangle whose area is 24 square feet; what are the sides of this triangle? The area of the triangle ABC=ABX ABC=96, also having AB and BC, AC may be found=20. Now it is evident that the triangles ABC and ADE are similar, and since the areas of sim, a's are as the squares of their like sides, we have, Area ABC: area ADE :: AC? : AE Area ABC : area ADE :: AB? :-AD2 %. A gentleman in his yard has a circular grass-plot, the diameter of which is 25 yards. Query the length of th string (nat would describe a circle to contain nine times a niuch. Ans. 37.5 yards. 4. Suppose a ladder 100 feet long, placed against a perpendicular wall 100 feet high, how far would the top of the ladder move down the wall by pulling out the bottom thereof 10 feet? Ans. .5012563, 5. *There is a circular pond whose area is 50284 square feet, in the middle of which stood a pole 100 feet high : now the pole having been broken, it was observed that the top just struck the brink of the pond; what is the height of the pole? Ans. 41.9968. 6. tIn a level garden there are two lofty firs, having their tops ornamented with gilt balls; one is 100 feet high, the other 80, and they are 120 feet distant at the bottom; now the owner wants to place a fountain in a right line between the trees, to be equally distant from the top of each; what will be its distance from the bottom of each tree, and also from each of the balls ? From the bottom of the lower tree 75 feet. Ans. From the bottom of the higher tree 45 feet: From each ball 109.6585 feet. * This problem may be constructed by forming a right angled triangle, having the radius of the circle for the base, and the length of the pole for the perpendicular; and erecting a perpendicular on th middle of the hypothenuse to cut the perpendicular of the triangle this will determine the place where the pole was broken. † The figure to this question is thus constructed. Draw AC=120 the distance of the trees at the bottom, and erect the perpendiculars 7. A person wishes to inclose bac. 1ro. i2po. in a tri. angle similar to a small triangle whose sides are 9, 8, and 6 perches respectively; required the sides of the triangle. Ans. 59.029, 52.47, and 39.353 perches. 8. Required the sides of an isosceles triangle, containing 6ac. Oro. 12per. and whose base is 72 perches. Ans. 45 perches each. AE=height of the lower tree, and CD=the higher. Join ED, and from the middle of it draw the perpendicular GF, and F will represent the place of the fountain. Join EF and DF, and draw EI parallel to AC, and GB parallel to DC; then the triangles EID and SBF being similar, the calculation is evident. OF THE CONIC SECTIONS. DEFINITIONS. 1. The conic sections are such plain figures as are formed by the cutting of a cone. 2. *A cone is a solid described by the revolution of a right-angled triangle about one of its legs, which remains fixed. 3. The axis of the cone is the right line about which the triangle revolves. * This is Euclid's definition of a cone, and is that which is gene. rally best understood by a learner ; but the following one is more general. Conceive the right line CB to move upon the fixed point C as a centre, and so as continually to touch the circumference of the circle AB, placed in any position, except in that of a plane which passes through the said point; and then that part of the line which is inter. cepted between the fixed point and the periphery of the circle will generate the convex superficies of a cone. 4. The base of a cone is the circle whicn is cescribeu dy she revolving leg of the triangle. 5. If a cone be cut through the vertex, by a plane which also cuts the base, the section will be a triangle. 6. If a cone be cui into two parts, by a plane parallel to the base, the section will be a circle. 7. If a cone be cut by a plane which passes through its !wo slant sides in an oblique direction, the section will be an ellipsis. 8. The longest straight line that can be drawn in an el. *psis is called the transverse axis; and a line drawn per pendicular to the transverse axis, passing through the centre of the ellipse, and terminated both ways by the circumfer. epce, is culled the conjugate axis. |