ORIGINAL CONSTRUCTIONS 1. To construct a fourth proportional to lines that are exactly 3 in., 5 in., and 6 in. long. How long should this constructed line be? 2. To construct a mean proportional between lines that are exactly 4 in. and 9 in. How long should this constructed line be? 3. Will a fourth proportional to three lines 5 in., 8 in., and 10 in., be the same length as a fourth proportional to 5 in., 10 in., and 8 in.? to 8 in., 10 in., and 5 in.? to 10 in., 5 in., and 8 in. ? 4. To construct a third proportional to lines that are exactly 3 in. and 6 in. long. 5. To produce a given line AB to point P, such that AB: AP = 3 : 5. [Divide AB into three equal parts, etc.] 6. To divide a line 8 in. long into two parts in the ratio of 5:7. [Divide the given line into 12 equal parts.] 7. To solve No. 6 by constructing a triangle. [See 308.] 8. To divide one side of a triangle into segments proportional to the other two sides. 9. To divide one side of a triangle externally into segments proportional to the other sides. 10. To construct two straight lines having given their sum and ratio. [Consult No. 6.] 11. To construct two straight lines having given their difference and ratio. [Consult No. 5.] 12. To construct a triangle similar to a given triangle and having a given perimeter. [First, use 358.] 13. To construct a right triangle having given its perimeter and an acute angle. [Constr. a rt. ▲ having the given acute 2. Etc.] 14. To construct a triangle having given its perimeter and two angles. [Constr. a ▲ having the two given. Etc.] 15. To construct a triangle similar to a given triangle and having a given altitude. 16. To construct a rectangle similar to a given rectangle and having a given base. 17. To construct a rectangle similar to a given rectangle and having a given perimeter. 18. To construct a parallelogram similar to a given parallelogram and having a given base. 19. To construct a parallelogram similar to a given parallelogram and having a given perimeter. 20. To construct a parallelogram similar to a given parallelogram and having a given altitude. 21. Three lines meet at a point; it is required to draw through a given point another line, which will be terminated by the outer two and be bisected by the inner one. Construction: From E on BD draw ||s. Etc. Through P draw RT || to GF. Statement: RS = ST. T C 22. To inscribe in a given circle a triangle similar to a given triangle. Construction Circumscribe a O about the given ▲; draw radii to the vertices; at center of given ○ construct 3 = to the other central angles. 23. To circumscribe about a given circle a triangle similar to a given triangle. Construction: First, inscribe a ▲ similar to the given A. 24. To construct a right triangle, having given one leg and its projection upon the hypotenuse. 25. To inscribe a square in a given semicircle. Construction: At Berect BD 1 to AB and = AB; draw DC, meeting O at R; draw RU || to BD. Etc. Statement: (?). Proof: RSTU is a rectangle (?). ACRU is similar to ▲ CBD (?). ..CU: CB RU: BD (?). But CB = } BD (?). :. CU = } RU (?). Etc. 26. To inscribe in a given semicircle, a rectangle similar to a given rectangle. Construction: From the midpoint of the base R draw line to one of the opposite vertices. At given center construct an = the ≤ at the midpoint. Proceed as in No. 25. Proof First, prove a pair of ▲ similar. Etc. 27. To inscribe a square in a given triangle. Construction: Draw altitude AD; con 28. To inscribe in a given triangle a rectangle similar to a given rectangle. Construction: Draw the altitude. On this construct a rectangle similar to the given rectangle. Proceed as in No. 27. 29. To construct a circle which shall pass through two given points and touch a given line. Given Points A and B; line CD. Construction: Draw line AB meeting CD R at P. Construct a mean proportional between PA and PB (by 361). On PD take PR = this mean. Erect OR to CD at R, meeting 1 bi sector of AB at 0. Use O as center, etc. 30. To construct a line = √2 in. [Diag. of square whose side is 1 in.] 31. To construct a line = √5 in. [Hyp. of a rt. ▲, whose legs are 1 in. and 2 in. respectively.] 32. To divide a line into segments in the ratio of 1: √2. 33. To divide a line into segments in the ratio of 1: √5. 34. To construct a line x, if x = ab с and a, b, c are lengths of three given lines. [That is, to construct x, if c: a = b : x (291).] 38. To construct a line x, if x = √a2 - b2. [a + b : x = x : a − b.] 43. To construct a line = Va2 + b2, if a and b are given lines. BOOK IV AREAS 366. The unit of surface is a square whose sides are each a unit of length. Familiar units of surface are the square inch, the square foot, the square meter, etc. 367. The area of a surface is the number of units of surface it contains. The area of a surface is the ratio of that surface to the unit of surface. A UNIT OF UNIT OF LENGTH Equivalent (~) figures are figures having equal areas. NOTE. It is often convenient to speak of "triangle," "rectangle," etc., when one really means "the area of a triangle," or "the area of a rectangle," etc. 368. THEOREM. If two rectangles have equal altitudes, they are to each other as their bases. Proof: I. If AB and EF are commensurable. There exists a common unit of measure of AB and EF (238). Suppose this unit is contained 3 times in AB and 5 times in EF. Hence, AB EF = 3 : 5 (Ax. 3). Draw lines through these points of division, to the bases. These will divide rectangle AC into three parts and EG into 5 parts, and all of these eight parts are equal (?) (140). Hence, AC: EG 3:5 (?). = .. AC : EG = AB : EF (Ax. 1). Q.E.D. Indefinitely increase the number of equal parts of AB; that is, indefinitely decrease each part, or the unit or divisor. Hence the remainder, RF, will be indefinitely decreased (?). That is, RF will approach zero as a limit, and, RFGS will approach zero as a limit. Hence, ER will approach EF as a limit (?), and ES will approach EG as a limit (?). Therefore, will approach AC AC as a limit (?), ES EG 369. THEOREM. Two rectangles having equal bases are to each other as their altitudes. (Explain.) 370. THEOREM. Any two rectangles are to each other as the products of their bases by their altitudes. a A X B b Given: Rectangles A and B whose altitudes are a and c and bases b and d respectively. |