Ex. 736. The perimeter and the diagonal. To construct a rhombus, having given : Ex. 737. The two diagonals. Ex. 738. The perimeter and one diagonal. Ex. 740. The altitude and the base. Ex. 741. The altitude and one angle. To construct a parallelogram, having given : Ex. 742. Two adjacent sides and one altitude. Ex. 743. Two adjacent sides and an angle. Ex. 744. One side and two diagonals. Ex. 745. One side, one angle, and one diagonal. Ex. 746. The diagonals and the angle formed by the diagonals. 269. In the analysis of a problem relating to a trapezoid, draw a line through one vertex, A, either parallel to the opposite arm, DC, or B parallel to a diagonal, DB. To construct a trapezoid, having given : Ex. 747. The four sides. Ex. 748. The bases and the base angles. Ex. 749. The bases, another side, and one base angle. Ex. 750. The bases and the diagonals. Ex. 751. One base, the diagonals, and the angle formed by the diagonals. Ex. 752. To draw a common external tangent to two given circles. Ex. 753. To draw a common internal tangent to two given circles. Ex. 754. About a given circle, to circumscribe a triangle, having given the angles. Ex. 755. Find the locus of the mid-points of the secants that pass through a given point without a circle. Ex. 756. In a given circle, to inscribe a triangle, having given the angles. * Ex. 757. From a given point in a circumference, to draw a chord that is bisected by a given chord. Ex. 758. Given a point, A, between a circumference and a straight line. Through A, to draw a line terminated by the circumference and the given line, and bisected in A. Ex. 759. Given two points, A and B, on the same side of a line, CD. To find a point, X, in CD, such that ▲ AXC = ▲ BXD. [See practical problems 44–53, pp. 293 and 294.] *A BOOK III PROPORTION. SIMILAR POLYGONS 270. DEF. A proportion is a statement of the equality of Hence, if a hypothesis states a : b = c:d, we may in the proof employ 271. DEF. The first and the fourth terms of a proportion are called the extremes, the second and the third, the means. 272. DEF. The first and the third terms are called the antecedents, the second and the fourth the consequents. Thus, in the proportion, a: b = c:d, a and d are the extremes, b and c the means, a and c the antecedents, and b and d the consequents. 273. DEF. When the means of a proportion are equal, either mean is said to be the mean proportional between the first and the last terms, and the last term is said to be the third proportional to the first and the second terms. Thus, in the proportion, a : b = b:c, b is the mean proportional between a and c, and c is the third proportional to a and b. 274. DEF. The last term is the fourth proportional to the first three. Thus, in the proportion, a : b = c : d, d is the fourth proportional to a, b, and c. 275. The two terms of a ratio must be either quantities of the same denomination, or the quantities must be represented by their numerical measures only. PROPOSITION I. THEOREM 276. In any proportion, the product of the means is equal to the product of the extremes. ad=bc. Clearing of fractions, i.e. multiplying both members by bd, (Ax. 7.) Q. E. D. 277. COR. If any three terms of a proportion are respectively equal to the three corresponding terms of another proportion, the remaining terms are equal. 278. NOTE. The product of two quantities, in Geometry, means the product of the numerical measures of the quantities. 279. If the product of two numbers is equal to the product of two other numbers, either two may be made the means, and the other two the extremes, of a proportion. Ex. 763. If ab = mn, find all possible proportions consisting of a, b, m, and n. Ex. 764. Form two proportions commencing with 3 from the equation 3 x 10 = 5 × 6. Ex. 765. If ab = xy, form two proportions commencing with b. Ex. 766. Find the ratio of x: y, if 280. A mean proportional between two quantities is equal to the square root of their product. |