CONSTRUCTION.-[(1.) By Prop. 20, taking a straight line BP, terminated in B, and making an angle PBG = with AB produced equal to D, draw a triangle BPQ = C. Bisect BP in E; and (2.)-] Make the parallelogram BEFG * = C, and having the angle EBG D B M H Complete the parallelogram AE. Join HB, and produce it to meet FG produced in K. Complete the parallelogram FL. Then BL is the required parallelogram. Produce EB to meet LK in M. = PROOF. The complement BL C. And the angle ABM EBG = D. Proposition 45.-Problem. To draw a parallelogram equal to any rectilinear figure (ABCD...), and having an angle equal to a given angle (E). * It is not necessary to write out the bracketed part of the construction, which is inserted here as an exercise. CONSTRUCTION.-Draw a parallelogram FGKL equal to the triangle ABC, and having an angle FGK = E. F H Y M E To HK apply a parallelogram HM = ACD, having an angle HKM = E. And so on, for as many triangles as ABCD... consists of. Then FX is the required parallelogram. = FGK = E = PROOF. Since the angle FHK HKM, therefore FH is parallel to KM. In a similar manner it may be shown that every part of GX is parallel to FH; and that every part of FY is parallel to GK. Therefore FY and GX are straight lines, and parallels. Hence FX is a parallelogram; and it is equal to the figure ABCD..., and has the angle FGX E Proposition 46.-Problem. = E. To draw a square on a given straight line (AB). = CONSTRUCTION.-Draw AC, BD at JD right angles. Cut off AE, BF AB; F and join EF. Then AF is a square. PROOF.-AF is equilateral. And since A and B are right angles, AE, BF are parallel (Prop. 27) as well as equal. Therefore AB, EF are parallel (Prop. 33); and AEF, EFB are right angles. Hence AF is a square (Def. 25). COR.-Hence it is clear that if one angle of a parallelogram is a right angle, so is each of the others. Proposition 47.-Theorem. The square on the hypotenuse of a right-angled triangle is equal to the squares on the sides. If C is the right angle in ABC, the square on AB the = squares on AC, CB. GCB, ACB are right F straight line; and it is parallel to BH. H Then, since the triangle CBD and the parallelogram BK are on the same base, and between the same parallels BD, CK, BK is double of CBD. Again, for a like reason, BG is double of ABH. = CBA (being a right angle), therefore the angle ABH the angle DBC; and the sides AB, BH DB, BC. = AK. Wherefore the triangle ABH = the triangle DBC; and consequently their doubles GB and BK are equal. Similarly we may show that FA Therefore AK and BK, that is, the square on AB, squares on AC and CB. the = Proposition 48.-Theorem. = If the square on one side of a triangle is equal to the squares on the other sides, the angle contained by these latter sides is a right angle. == the squares If the square on AB on AC, CB, ABC is a right angle. Draw CD at right angles to AC, and make CD CB. Join AD. Then ACD is a right-angled triangle, and the square on AD AC, CD. = the squares on Therefore the squares on AC, CD on AC, CB. That is, the square on AD the square on AB; or AD AB. Hence, in the triangles ACD, ACB, we have AC, CD AC, CB, and AD = = AB; Therefore, the angle ACD = the angle ACB. But ACB is a right angle; therefore, ACB is a right angle. ALGEBRA. EXPLANATION OF SYMBOLS. 1. Algebra is an abbreviation of Arithmetic, by means of symbols, contracted methods, and the representation of unknown quantities. 2. The letters of the alphabet are used to represent either abstract numbers or concrete quantities. The last letters of the alphabet, especially x, y, and z, usually denote quantities whose numerical value is not known. (Compare the use of x in arithmetical proportion.) 66 minus; b. 3. The sign of addition is +, "plus;" thus, a + b. The sign of subtraction is thus, a When it is uncertain which of two terms is the greatest, their difference is expressed thus, ab. Multiplication is expressed by x, by a point, or by the absence of signs; thus, a x b, a. b, or ab. Division is expressed thus, a ÷ b, or %.. 4. The sign of equality is =; .. signifies "therefore;" "because;">, "greater than; <,"less than." 5. When a term is multiplied by itself, it is said to be raised to a power. Thus a × a = second power of a, * So a xa x a written a2; and inversely, a2 = a. = a; and as = a. The square root of a, or Ja, is also written a1, or 1 a ; so a is written a or The * Called " a squared." The powers are also called dimensions of the base. |