3 Ex. 60. Two angles of an inscribed quadrilateral are 90° and 60°, respectively. Find the remaining angles and inscribe the quadrilateral in a given circle. Can more than one such quadrilateral be inscribed ? One side of a triangle is three-fourths another, and threefifths the third side. The perimeter of the triangle is 48. Construct the triangle. Ex. 62. A given point lies within, without, or on the circumference of, a circle. With the given point as centre, describe a circumference passing through the extremities of a given diameter of the circle. Ex. 63. We define the angle between two intersecting curves as thê angle formed by tangents to the curves at their point of intersection. With a point outside a given circle as a centre, describe an arc which shall intersect the given circumference at right angles. BOOK III PROPORTION.—SIMILAR POLYGONS DEFINITIONS 210. A Proportion is a statement that two ratios are equal. 211. The statement that the ratio of a to b is equal to the ratio of c to d is written 810 C 212. In the proportion == a b we call a the first term, b the second term, c the third term, and d the fourth term. 213. We call the first and fourth terms the extremes, and the second and third terms the means. We call the first and third terms the antecedents, and the second and fourth terms the consequents. Thus, in the above proportion a and d are the extremes, b and c the means, a and c the antecedents, and b and d the consequents. 214. If the means of a proportion are equal, either mean is called a mean proportional between the first and last terms, and the last term a third proportional to the first and second terms. b Thus, in the proportion 2, b is a mean proportional be α b tween a and c, and c a third proportional to a and b. 215. In the proportion = d is called a fourth propor tional to a, b, and c. ď 216. In any proportion, the product of the extremes is equal to Proof. Multiplying both members of equation (1) by bd, (1) That is, the mean proportional between two numbers is equal to the square root of their product. PROP. II. THEOREM 218. (Converse of Prop. I.) If the product of two numbers is equal to the product of two others, one pair may be made the extremes, and the other pair the means, of a proportion. Proof. Dividing both members of (1) by bd, (1) 219. In any proportion, the terms are in proportion by ALTERNATION; that is, the first term is to the third as the second term is to the fourth. 220. In any proportion, the terms are in proportion by INVERSION; that is, the second term is to the first as the fourth term is to the third. 221. In any proportion, the terms are in proportion by COMPOSITION; that is, the sum of the first two terms is to the first term as the sum of the last two terms is to the third term. Adding both members of this equation to ac, ac+adac + be, or a(c + d) = c(a+b). (?) (2) 222. In any proportion, the terms are in proportion by DIVISION; that is, the difference of the first two terms is to the first term as the difference of the last two terms is to the third term. Subtracting both members from ac, ac― ad ac― bc, or a(cd) = c(a — b). 223. In any proportion, the terms are in proportion by COMPOSITION AND DIVISION; that is, the sum of the first two terms is to their difference as the sum of the last two terms is to their difference. Proof. Dividing equation (2), § 221, by equation (2), § 222, Ex. 1. What is the ratio of 50 cents to $1? of hour to 1 day? of 150 rods to 1 mile ? Ex. 2. If x is a fourth proportional to a, b, c, find x. Ex. 3. Apply composition and division to the following: |