Q. LXXXI. To bisect a triangle by a line drawn parallel to one of its Fig. 367. sides. Let ABC be the given triangle to be bisected by a line parallel to its side AB. On BC describe a semicircle; bisect BC in O, and draw the perpendicular OD; join CD; and, with C as a centre and radius CD, describe a circle cutting CB in E; draw EF parallel to AB; EF bisects the triangle. (215.) Q. LXXXII. If the sides AB, CB, of the triangle ABC inscribed in the Fig. 368. segment ABC, be produced to meet CE, AD, lines drawn from the extremities of the base, forming with it angles equal to the angle of the segment, the rectangle contained by these lines will be equal to the square described on the base. (202.) Q. LXXXIII. If a rectangular parallelogram DBEF is inscribed in a right Fig. 369. angled triangle ABC, and they have the right angle common, the rectangle contained by the segments of the hypothenuse AF, FC, is equal to the sum of the rectangles contained by the segments of the sides about the right angle AD, DB, and BE, EC. Q. LXXXIV. If an isosceles triangle ABC be inscribed in a circle, and Fig. 370. from the vertical angle A, a line AD be drawn meeting the circumference and the base, either equal side AB, AC, is a mean proportional between DA and AE. Q. LXXXV. If any triangle ABC be inscribed in a circle, and from the Fig. 371. vertex B a line be drawn parallel to a tangent at either extremity of the base, this line BD will be a fourth proportional to the base and two sides; that is, AC:AB:: BC: BD. Q. LXXXVI. If, from O the centre of a circle, a line be drawn to any Fig. 372. point C of the chord AB, the square of the line OC, together with the rectangle AC, CB, contained by the segments of the chord, will be equal to the square described on the radius. Through C draw DE perpendicular to OC. Q. LXXXVII. If, from any point D in the base or base produced of the Fig. 373. segment of a circle ABC, a line DE be drawn, making with AC an angle equal to the angle in the segment, and meeting any line AB drawn from the extremity A, and cutting the circumference, the rectangle EA, AB, contained by this line and the part within the segment, is always of the same magnitude. The rectangle AC, AD, is invariable. Q. LXXXVIII. If, from any point D in the diameter AC of a semicircle Fig. 374. ABC, a perpendicular DF be drawn, and from the extremities of the diameter lines AB, CB, be drawn meeting the perpendicular in the points E and F, the rectangle contained by the parts DE, DF, which they cut off from the perpendicular, will be equal to the rectangle contained by the segments of the diameter AD, DC. Q. LXXXIX. If the diameter of a semicircle be divided into any num- Fig. 375. ber of parts, and on them semicircles be described, their circumferences will together be equal to the circumference of the given semicircle. Tet AB, the diameter of the semicircle ACB, be divided into any number of parts in the points D, E, and on AD, DE, EB, let semicircles be described: their circumferences are together equal to ACB. (287.) Q. XC. The difference of level for one mile being found by observation to be 8 inches nearly, what is the diameter of the earth? The tangent being the apparent level, the arc the true level, the part of the secant without the circumference will be the difference of level. (225. 187.) Ans. 7920 miles very nearly. Geom. 30 Q. XCI. The radius of the earth at the equator being found to be 3963,2 miles, what is the circumference of the equator? (294.) Ans. 24901,578 miles. Q. XCII. The radius of the earth at the poles being found to be 3950,4 miles, what is the circumference of a meridian, on the supposition that it is a circle? (294.) Ans. 24821,153 miles. Q. XCIII. The mean diameter of the earth, considered as a sphere, being 7912 miles, what is the circumference of a great circle? Ans. 24856 miles. 7912 Q. XCIV. The earth being a sphere whose diameter is 7912 miles, what is the distance round it on the parallel of latitude 42° 23′ 28′′ N., the radius of the small circle being 2922 miles? Ans. 18359,5 miles. Q. XCV. What is the length of a degree on this parallel? Ans. 51 miles very nearly. Q. XCVI. What is the area of the circle of which the equator is the circumference? Ans. 49344812,4 square miles. Q. XCVII. What is the area of a sector whose arc is 90°, the diameter being 20 feet? Ans. 78,54 square feet. Q. XCVIII. What is the area of a sector whose arc is 120°, and whose radius is 100 feet? Ans. 10472 square feet. Q. XCIX. What is the ratio and what the difference of the areas of two circles, the radius of one being 64 feet, and that of the other 256 feet? Ans. Ratio 1: 16. dif. of areas, 193019,9040 square feet. Q. C. What is the area of the ring enclosed between the circumferences of two concentric circles whose diameters are 10 feet and 6 feet? Ans. 50,2656 square feet. CI. What is the area of a segment whose radius is 10 feet, and its arc 90°? CIII. Required the area of a segment AGB (fig. 51) whose chord AB is 20 feet, its height DG 2 feet, and its are 15,243 (1917, to and the diame Ans. 26,881 square feet ter.) Q. CIV. Required the area of a similar segment whose radius is 52 feet. Ans. 107,524 square feet. Q. CV. How many solid feet in a square prism whose altitude is 5 feet, and each side of its base 1 feet? (406.) Ans. 97 solid feet. Q. CVI. What is the solidity of a quadrangular pyramid each side of whose base is 30 feet, the altitude of each face being 25 feet? (416.) Ans. 6000 solid feet. Q. CVII. What is the solidity of the frustum of a quadrangular pyramid, the side of the inferior base being 15 inches, that of the superior 6 inches, and the altitude 24 feet? (422.) Ans. 19,5 solid feet. Q. CVIII. The dimensions of the great pyramid Geeza in Egypt, as derived from the accurate measurement of M. Coutelle, are as follows:-Side of the base, 763,62 feet; entire altitude, 476,01; altitude of the frustum, 456,43 feet. Required the area of the base, the whole surface and the solid contents of the frustum. Ans. Area of the base, 583115,5 square feet. CIX. What is the solidity of a cylinder whose altitude and the circumference of whose base are each 20 feet? (516.) Ans. 636,62 solid feet. Q. CX. What is the convex surface of a right triangular prism whose altitude is 20 feet, and each side of the base 18 inches? (520.) Ans. 90 square feet. Q. CXI. What is the surface of a cube whose side is 20 feet? Ans. 2400 square feet. Q. CXII. What is the convex surface of a cylinder whose altitude is 20 feet, and the diameter of its base 2 feet? (523.) Ans. 125,664 square feet. Q. CXIII. What is the whole surface of a cylinder whose altitude is 10 feet, and the circumference of its base 3 feet? Ans. 31,4324 square feet. Q. CXIV. What is the convex surface of a quadrangular pyramid whose altitude is 20 feet, and each side of the base 30 feet? ARs. 1500 square feet. Q. CXV. What is the convex surface of a cone whose side is 20 feet, and the circumference of its base 9 feet? (528.) Ans. 90 square feet. Q. CXVI. How many square feet in the surface of the frustum of a quadrangular pyramid, the altitude of each face being 10 feet, each side of the inferior base 3 feet 4 inches, and of the superior 2 feet 2 inches? Ans. 110 square feet. Q. CXVII. The earth being supposed a perfect sphere, whose diameter is 7912 miles, what is its surface? (535.) 196660672 square miles. Ans.. Q. CXVIII. What is the convex surface of the frigid zone of the earth, the altitude of this zone being 327 miles? (538.) Ans. 8127912 square miles. Q. CXIX. What is the convex surface of the torrid zone, the altitude being 3150,6 miles? Ans. 78311313,6 square miles, Q. CXX. What is the convex surface of either temperate zone, the altitude being 2053,7 miles? Ans. 51046767,2 square miles. Q. CXXI. What is the solidity of the earth? (546.) Ans. 259328406532 solid miles nearly. Q. CXXII. What is the solidity of the spherical segments of which the frigid zones are the convex surfaces, the altitude of each segment being 327 miles, and the radius of the base 1575,28 miles? Ans. 1282921583 solid miles nearly. ૨. CXXIII. What is the solidity of the spherical segments of which the temperate zones are the convex surfaces, the radius of the superior base being 1575,28 miles, that of the inferior 3628,86 miles, and the altitude 2053,7 miles? Ans. 55021192817 solid miles nearly. ૨. CXXIV. What is the solidity of the spherical segment of which the torrid zone is the convex surface, the radii of the bases being 3628,86 miles, and its altitude 3150,6 ? Ans. 146715018499 solid miles nearly. |