287. PROPOSITION XXI.--Problem. Given any polygon, A B C D E, and the unequal lines m and n. Construct a polygon similar to A B C D E and having the ratio to it of m to n. If you fail, see "hint" below. [Hint.-How do similar polygons vary? Can you construct a square whose ratio to the square of a side of A B C D E shall equal mn?] 295. Construct a ▲ equivalent to a given polygon having given the base and median to the base of the A. 288. PROPOSITION XXII.-Problem. Given two dissimilar polygons, A and B. It is required to construct a polygon similar to A and equivalent to B. [Hint.--(1) Assume the problem solved. Then we have given dissimilar polygons A and B, and the required polygon X, similar to A and equivalent to B. Let a = a side of A, and x = a side of X, the required polygon. Review the propositions about similar polygons. Form proportions. You wish to discover a method to find the length of what line? (With it known you can construct X.)] [Hint.-(2) A: X = a2 : x2. [?] How many unknown terms in the above proportion? Put for one of the unknown terms a known term. Are all the magnitudes in the above proportion of the same kind? Get another proportion from this, in which the magnitudes are not areas. What kind of magnitude is A? What does the VA mean? Can you construct it?] (2) Having one diagonal equal to the long diagonal of the rhomboid. 297. How does the radius of an inscribed circle in an equilateral compare with the radius of the circumscribed circle? 298. Construct a similar to one of two given dissimilar As and equivalent to their difference. Discuss. 299. Construct an isosceles A equivalent to a given ▲, having given one of the equal sides. Discuss. 300. Draw a line parallel to a side of an equilateral A which will divide it into two equivalent parts. the area of the [Hint.-(1) Suppose the problem solved, and then compare cut off with that of the original. How Form the proportion, etc.] do these As vary? If you still fail, consult figure and further "hint." 2:1 = a2: x2. [?] But T2 T; [?] = a2x2; (What is the unit of measure here?) And V2:1a: x. (What is the unit of measure here?) Construct the V2 where the unit is given. Construct the converse. 301. Draw a line parallel to the base of a given ▲ dividing it into two equivalent parts. 302. Draw a line through a given point in the side of a A dividing it into two equivalent parts. [Hint.-Draw a scalene different positions in the sides. and solve, fixing the point in P When P is joined to opposite vertex, is the A divided into equivalent As? Why not? Where must P be in order that the As shall be equivalent? Fix this point. Call it m, join P and m to opposite vertex. The line from m cuts off half. How much does the line through P lack of cutting off half? How then can you draw a line through P which will cut off in addition to already cut off a equivalent to A lacking?] 303. Draw a line through any point in the side of a parallelogram dividing it into two equivalent parts. When will the parts be As? trapezoids? MOULDING OF POLYGONS. $283 may be stated as follows: Mould a polygon into an equivalent A. Note the figure carefully. A CF is equivalent to AACB. Why? In A A C B consider A C the base and let the vertex B be moved parallel to the base A C. Suppose we start to move B toward F, but stop at intermediate points, x, y, z, etc. Why are As BAC, XA C, y AC, z AC ......and F A C equivalent? Are any of these As isosceles? right? equilateral? 304. I. Redraw A BAC ($283) and mould it into (1) an isosceles A; (2) a right /\; (3) a different isosceles ▲ from (1). What is this base here? (4) Still a different isosceles ▲ from (1) or 3. What is now the base? II. Now redraw A B A C and take A B the base. What is the vertex? Mould B A C into the following As, using A B the base: (1) isosceles; (2) right. Can you mould it into other isosceles As while A B is considered the base? Why? III. Answer questions as asked in II. using B C the base. IV. Discuss the possibilities of moulding a scalene A into different shaped equivalent isosceles As. 305. Divide any quadrilateral into two equivalent parts by drawing a line through any given point in any one of its sides. [Hint.-Mould the quadrilateral into a A.] Discuss. 306. Draw two scalenes having unequal altitudes. Draw an isosceles equivalent to their sum. [Hint.-(1) Mould each into a right A. Raise the altitude of T' until its altitude equals that of T. T is equivalent to A BC. [?] T' is equivalent to MNO. [?] To solve: (2) Draw M R and then draw through O a line parallel to M R, cutting M N at S. Draw S R. SRN is equivalent to Why? M N O, which is equivalent to T. (3) Now add the As by adding their bases and mould into the required isosceles A. 307. Draw three scalene As of unequal altitudes, and (1) Mould into a rt. ▲ equivalent to their sum; (2) Draw a rectangle equivalent to their sum; (3) Draw a square equivalent to their sum. 308. (1) Draw a quadrilateral and mould it into an equivalent A. Now mould the A into an equivalent quadrilateral. Discuss the data necessary to mould the into the original quadrilateral. (2) Draw a square ‡ as large as a given square. 309. Draw an irregular pentagon and then draw a square having the area of the pentagon. 310. Mould a scalene A (1) into a "kite" trapezium; (2) into an "arrow" trapezium. 311. Construct a similar to a given A and having twice the area. 312. Draw a circle having twice the area of a given circle. 313. Draw a square having the area of a given square. 314. Draw a circle having half the area of a given circle. 315. Draw an irregular polygon; then draw a similar polygon having 3% the area. 316. Draw a "kite" trapezium and mould it into an equivalent A. 317. Draw an irregular polygon having two re-entrant Zs. Mould it into an equivalent square. 318. Divide a given line into 2 equal parts. 319. Draw two similar but unequal rectangles. Then draw a rectangle similar and equivalent to their sum. |