= Ex. 659. In the diagram at the right, if AB = 15, AC = 12, AD = 10, and AE = 8, prove DE parallel to BC. Ex. 660. If AB = 50, DB = 15, and EC 12, is DE parallel to BC? = AE = 28, DB = 5, Ex. 662. If DE || BC, AB= 80, AD = 25, and EC = 4, find AC. SIMILAR POLYGONS B 417. Def. If the angles of one polygon, taken in order, are equal respectively to those of another, taken in order, the polygons are said to be mutually equiangular. The pairs of equal angles in the two polygons, taken in order, are called homologous angles of the two polygons. 418. Def. If the sides of one polygon, taken in order as antecedents, form a series of equal ratios with the sides of another polygon, taken in order as consequents, the polygons are said to have their sides proportional. Thus, in the accompanying figure, if a : 1 = b : m = c : n = d : oe: p, the two polygons have their sides proportional. The lines forming any ratio are called homologous lines of the two polygons, and the ratio of two such lines is called the ratio of similitude of the polygons. 419. Def. Two polygons are similar if they are mutually equiangular and if their sides are proportional, PROPOSITION XIV. THEOREM 420. Two triangles which are mutually equiangular are similar. = = F Given ABC and DEF with ZALD, LB LE, and LC LF. To prove ▲ ABC ~ ▲ DEF. ARGUMENT 1. Place A DEF on ▲ ABC so that D shall coincide with A, DE falling on AB Represent ▲ DEF in .. HK BC. and DF on AC. 3. 4. 5. AB AC = AH ᎪᏦ AB AC = DE DF 6. By placing ▲ DEF on ▲ ABC so that LE shall coincide with B, it may 421. Cor. I. If two triangles have two angles of one equal respectively to two angles of the other, the triangles are similar. 422. Cor. II. Two right triangles are similar if an acute angle of one is equal to an acute angle of the other. 423. Cor. III. If a line is drawn parallel to any side of a triangle, this line, with the other two sides, forms a triangle which is similar to the given triangle. Ex. 663. Upon a given line as base construct a triangle similar to a given triangle. Ex. 664. Draw a triangle ABC. Estimate the lengths of its sides. Draw a second triangle DEF similar to ABC and having DE equal to two thirds of AB. DF and EF. Compute Ex. 665. Any two altitudes of a triangle are to each other inversely as the sides to which they are drawn. Ex. 667. If CA is found to be 64 feet; DF, 16 feet; ED, 10 feet; what is the height of the tree? Ex. 668. To find the distance across a river from A to B, a point C was located so that BC was perpendicular to AB at B. CD was then measured off 100 feet in length and perpendicular to BC at C. The line of sight from D to A intersected BC at E. By measurement CE was found to be 90 feet and EB 210 feet. What was the distance across the river ? A E B Ex. 669. Two isosceles triangles are similar if the vertex angle of one equals the vertex angle of the other, or if a base angle of one equals a base angle of the other. 424. It follows from the definition of similar polygons, § 419, and from Prop. XIV that: (1) Homologous angles of similar triangles are equal. (2) Homologous sides of similar triangles are proportional. (3) Homologous sides of similar triangles are the sides opposite equal angles. 425. Note. In case, therefore, it is desired to prove four lines proportional, try to find a pair of triangles each having two of the given lines as sides. If, then, these triangles can be proved similar, their homologous sides will be proportional. By marking with colored crayon the lines required in the proportion, the triangles can readily be found. If it is desired to prove the product of two lines equal to the product of two other lines, prove the four lines proportional by the method just suggested, then put the product of the extremes equal to the product of the means.* 426. Def. The length of a secant from an external point to a circle is the length of the segment included between the point and the second point of intersection of the secant and the circumference. Ex. 670. If two chords intersect within a circle, establish a proportionality among the segments of the chords. Place the product of the extremes equal to the product of the means, and state your result as a theorem. Ex. 671. If two secants are drawn from any given point to a circle, what are the segments of the secants? Does the theorem of Ex. 670 still hold with regard to them? Ex. 672. Rotate one of the secants of Ex. 671 about the point of intersection of the two until the rotating secant becomes a tangent. What are the segments of the secant which has become a tangent? Does the theorem of Ex. 670 still hold? Prove. *By the product of two lines is meant the product of their measure-numbers. This will be discussed again in Book IV. |