434. Def. A line is divided harmonically if it is divided internally and externally into segments whose ratios are numerically equal; thus, if line AC is divided internally at P and externally at Q so that the ratio of AP to PC is numerically equal to P C Q the ratio of 4Q to QC, AC is said to be divided harmonically. Ex. 689. The bisectors of the interior and exterior angles at any vertex of a triangle divide the opposite side harmonically. Ex. 690. Divide a given straight line harmonically in the ratio of 3 to 5; in the ratio of a to b, where a and b are given straight lines. PROPOSITION XX. THEOREM 435. In two similar triangles any two homologous altitudes have the same ratio as any two homologous sides. Given two similar A ABC and DEF, with two corresponding altitudes AH and DK. PROPOSITION XXI. THEOREM 436. If three or more straight lines drawn through a common point intersect two parallels, the corresponding segments of the parallels are proportional. Given lines PA, PB, PC, PD drawn through a common point P and intersecting the lines AD and A'D' at points A, B, C, D and A', B', C', D', respectively. Ex. 691. If three or more non-parallel straight lines intercept proportional segments on two parallels, they pass through a common point. Ex. 692. A man is riding in an automobile at the uniform rate of 30 miles an hour on one side of a road, while on a footpath on the other side a man is walking in the opposite direction. If the distance between the footpath and the auto track is 44 feet, and a tree 4 feet from the footpath continually hides the chauffeur from the pedestrian, does the pedestrian walk at a uniform rate? If so, at what rate does he walk? Ex. 693. Two sides of a triangle are 8 and 11, and the altitude upon the third side is 6. A similar triangle has the side homologous to 8 equal to 12. Compute as many parts of the second triangle as you can. Ex. 694. In two similar triangles, any two homologous bisectors are in the same ratio as any two homologous sides. Ex. 695. In two similar triangles, any two homologous medians are in the same ratio as any two homologous sides. 437. Drawing to Scale. Measure the top of your desk. Make a drawing on paper in which each line is as long as the corresponding line of your desk. Check your work by measuring the diagonal of your drawing, and the corresponding line of your desk. This is called drawing to scale. Map drawing is a common illustration of this principle. The scale of the drawing may be represented: (1) by saying, "Scale," or "Scale, 1 inch to 12 inches"; (2) by actually drawing the scale as indicated. A B 12 D 36 Ex. 696. Using the scale above, draw lines on paper to represent 24 inches; 3 feet 3 inches. Ex. 697. On the blackboard draw, to the scale above, a circle whose diameter is 28 feet. Ex. 698. The figure represents a farm drawn to the scale indicated. Find the cost of putting a fence around the farm, if the fencing costs $2.50 per rod. B PROPOSITION XXII. THEOREM 438. If two polygons are composed of the same number of triangles, similar each to each and similarly placed, the polygons are similar. Given polygons ABCDE and FGHIJ with ▲ ABC ~ ▲ FHI,▲ADE ~ ▲ FIJ. A ACD ~ To prove polygon ABCDE ~ ARGUMENT 1. In ▲ ABC and FGH, ≤B = ≤ G. 2. Also 1= Z 2. ▲ FGH, polygon FGHIJ. REASONS 3. In ▲ ACD and FHI, ≤3 = ≤ 4. 5. . BCD = GHI. 7. .. polygons ABCDE and FGHIJ are mutually equiangular. 8. In ▲ ABC and FGH, 9. In ACD and FHI, AB BC CA = FG GH HF 7. By proof. 8. § 424, 2. 9. § 424, 2. 10. § 424, 2. 11. § 54, 1. IJ JF IF IJ JF AB BC CD DE EA = = FG GH HI 12... polygon ABCDE 12. § 419. 439. Cor. Any two similar polygons may be divided into the same number of triangles similar each to each and similarly placed. PROPOSITION XXIII. PROBLEM 440. Upon a line homologous to a side of a given polygon, to construct a polygon similar to the given polygon. Given polygon AD and line MQ homol. to side AE. To construct, on MQ, a polygon ~ polygon AD. I. Construction 1. Draw all possible diagonals from A, as AC and AD. 2. At M, beginning with MQ as a side, construct 7, 8, and 10 equal to ≤ 4, and § 125. 11 equal to ≤5, and 9 equal respectively to 1, 2, and 3. § 125. 3. At Q, with MQ as a side, construct prolong side QP until it meets ML at P. 4. At P, with PM as a side, construct prolong side PO until it meets MR at 0. 5. At 0, with OM as a side, construct ≤12 equal to ≤6, and prolong side ON until it meets MF at N. 6. MP is the polygon required. § 125. § 125. II. Proof ARGUMENT. REASONS 1. Δ ADE ~ Δ MPQ, Δ ACD ~ Δ ΜΟΡ, and 1. § 421. ΔΑΒΕ ~ Δ ΜΝΟ. 2... polygon MP ~ polygon AD. Q.E.D. 2. § 438. |