ARGUMENT 1. Draw all possible diagonals from any REASONS 1. $ 54, 15.. 2. § 439. 3. § 503. 4. § 503. 5. § 503. 6. § 419. 7. § 54, 13. 8. § 54, 1. 9. § 401. Q.E.D. 10. § 309. 506. Cor. Two similar polygons are to each other as the squares of any two homologous diagonals. Ex. 879. If one square is double another, what is the ratio of their sides? Ex. 880. Divide a given hexagon into two equivalent parts so that one part shall be a hexagon similar to the given hexagon. Ex. 881. The areas of two similar rhombuses are to each other as the squares of their homologous diagonals. Ex. 882. One side of a polygon is 8 and its area is 120. The homologous side of a similar polygon is 12; find its area. PROPOSITION XII, THEOREM 507. The square described on the hypotenuse of a right triangle is equivalent to the sum of the squares described on the other two sides. Given rt. ▲ ABC, right-angled at C, and the squares described on its three sides. To prove square AD≈ square BF + square CH. ARGUMENT REASONS 1. From C draw CM AB, cutting AB at L 1. § 155. 10. ▲ CAK and rectangle AM have the same base AK and the same altitude, the 10. § 235. 14. But A CAK 15. .. ΔΗΑΒ. rectangle AM≈ square CH. 16. .. rectangle AM≈ square CH. 17. Likewise, by drawing CD and AE, it may be proved that rectangle LD≈ square BF. 18... rectangle AM+ rectangle LD square CH+ square BF. 19. .. square AD≈ square CH + square BF. Q.E.D. 13. § 489. 14. Arg. 9. 15. § 54, 1. 16. § 54, 7 a. 17. By steps similar to 5-16. 18. § 54, 2. 19. § 309. 508. Cor. I. The square described on either side of a right triangle is equivalent to the square described on the hypotenuse minus the square described on the other side. 509. Cor. II. If similar polygons are described on the three sides of a right triangle as homologous sides, the polygon described on the hypotenuse is equivalent to the sum of the polygons described on the other two sides. Given rt. ▲ ABC, right-angled at C, and let P, Q, and R be polygons described on a, b, and c, respectively, as homol. sides. To prove R≈ P + Q. ARGUMENT ONLY ~ Ex. 883. The square on the hypotenuse of an isosceles right triangle is equivalent to four times the triangle. 510. Historical Note. Prop. XII is usually known as the Pythago rean Proposition, because it was discovered by Pythagoras. The proof given here is that of Euclid (about 300 B.C.). Pythagoras (569-500 B.C.), one of the most famous mathematicians of antiquity, was born at Samos. He spent his early years of manhood studying under Thales and traveled in Asia Minor and Egypt and probably also in Babylon and India. He returned to Samos where he established a school that was not a great success. Later he went to Crotona in Southern Italy and there gained many adherents. He formed, with his closest followers, a secret society, the members of which possessed all things in common. They used as their badge the five-pointed star or pentagram which they knew how to construct and which they considered symbolical of health. They ate simple food and practiced severe discipline, having obedience, temperance, and purity as their ideals. The brotherhood regarded their leader with reverent esteem and attributed to him their most important discoveries, many of which were kept secret. Pythagoras knew something of incommensurable numbers and proved that the diagonal and the side of a square are incommensurable. The first man who propounded a theory of incommensurables is said to have suffered shipwreck on account of the sacrilege, since such numbers were thought to be symbolical of the Deity. Pythagoras, having incurred the hatred of his political opponents, was murdered by them, but his school was reestablished after his death and it flourished for over a hundred years. Ex. 884. Use the adjoining figure to prove the Pythagorean theorem. II II I Ex. 885. Construct a triangle equivalent to the sum of two given triangles. I B A Ex. 886. The figure represents a farm drawn to the scale indicated. Make accurate measurements and calculate approximately the number of acres in the farm. Ex. 887. A farm XYZW, in the form of a trapezium, has the following dimensions: XY = 60 rods, YZ = 70 rods, ZW =90 rods, WX = 100 rods, and XZ = - 66 rods. Draw a plot of the farm to the scale 1 inch = 40 rods, and calculate the area of the farm in acres. 511. Def. By the rectangle of two lines is meant the rectangle having these two lines as adjacent sides. Ex. 888. The square described on the sum of two lines is equivalent to the sum of the squares described on the lines plus twice their rectangle. Ex. 889. The square described on the difference of two lines is equivalent to the sum of the squares described on the lines diminished by twice their rectangle. HINT. Let AB and CB be the given lines. Ex. 890. The rectangle whose sides are the sum and difference respectively of two lines is equivalent to the difference of the squares described on the lines. HINT. Let AB and BC be the given lines. Write the three algebraic formulas corresponding to the last three exercises. Ex. 891. |