Given the three lines m, n, and p. Required to find the fourth proportional to m, n, and p. Construction. Draw two lines AX and AY containing any convenient angle. Then Proof. On AX take AB equal to m, and take BC equal to n. On AY take AD equal to p. Draw BD. From C draw CE | to BD, meeting AY at E. DE is the fourth proportional required. AB: BCAD: DE. § 233 Q.E.F. § 273 (If a line is drawn through two sides of a ▲ll to the third side, it divides the two sides proportionally.) Substituting m, n, and p for their equals AB, BC, and AD, we have m: n=p: DE. Ax. 9 Therefore DE is the fourth proportional to m, n, and p, by § 259. Q.E.D. 308. COROLLARY. To find the third proportional to two given lines. In the above proof take m, n, n as the given lines instead of m, n, p. PROPOSITION XXVII. PROBLEM 309. To find the mean proportional between two given Required to find the mean proportional between m and n. Construction. Draw any line AE, and on AE take AC equal to m, and CB equal to n. On AB as a diameter describe a semicircle. At C erect the 1 CH, meeting the circle at H. Then CH is the mean proportional between m and n. Proof. AC: CH=CH: CB. (The from any point on a circle to a diameter is the mean Substituting for AC and CB their equals m and n, we have m: CH=CH: n, by Ax. 9. § 228 Q.E. F. $297 Q.E.D. 310. Extreme and Mean Ratio. If a line is divided into two segments such that one segment is the mean proportional between the whole line and the other segment, the line is said to be divided in extreme and mean ratio. E.g. the line a is divided in extreme and mean ratio, if a segment x is found such that The division of a line in extreme and mean ratio is often called the Golden Section. 311. To divide a given line in extreme and mean ratio. C------ Given the line AB. Required to divide AB in extreme and mean ratio. Construction. At B erect a 1 BE equal to half of AB. § 228 From E as a center, with a radius equal to EB, describe a O. Draw AE, meeting the circle at F and G. On AB take AC equal to AF. On BA produced take AC' equal to AG. Then AB is divided internally at C and externally at C' in extreme and mean ratio. That is, AB: AC AC: CB, and AB: AC' = AC': C'B. Q. E. F. Proof. = AG-AB: AB= § 302 From AG: AB=AB: AF, PROPOSITION XXIX. PROBLEM 312. Upon a given line corresponding to a given side of a given polygon, to construct a polygon similar to the given polygon. Given the line A'B' and the polygon ABCDE. Β' Required to construct on A'B', corresponding to AB, α polygon similar to the polygon ABCDE. Construction. From A draw the diagonals AD and AC. From A'draw A'X, A'Y, and A'Z, making x', y', and z' equal respectively to x, y, and z. From B' draw a line, making LB' equal to ZB, and meeting A'X at C'. From C'draw a line, making D'C'B' equal to ▲ DCB, and meeting A'Y at D'. From D' draw a line, making E'D'C' equal to LEDC, and meeting A'Z at E'. Then A'B'C'D'E' is the required polygon. § 232 Q.E.F. Proof. The AABC and A'B'C', the ▲ ACD and A'C'D', and the AADE and A'D'E', are similar. § 286 Therefore the two polygons are similar, by § 293. Q.E.D. EXERCISE 48 = 3. 1. If a and b are two given lines, construct a line equal to x, where x = Vab. Consider the special case of a = 2, b 2. If m and n are two given lines, construct a line equal to x, where x = √2 mn. 3. Determine both by geometric construction and arithmetically the third proportional to the lines 1 in. and 2 in. 4. Determine both by geometric construction and arithmetically the third proportional to the lines 4 in. and 3 in. 5. Determine both by geometric construction and arithmetically the fourth proportional to the lines 1 in., 2 in., and 24 in. 6. Determine both by geometric construction and arithmetically the mean proportional between the lines 1.2 in. and 2.7 in. 7. Find geometrically the square root of 5. Measure the line and thus determine the approximate arithmetical value. 8. A map is drawn to the scale of 1 in. to 50 mi. How far apart are two places that are 23 in. apart on the map? 9. Find by geometric construction and arithmetically the third proportional to the two lines 1 in. and 2g in. 10. Divide a line 1 in. long in extreme and mean ratio. Measure the two segments and determine their lengths to the nearest sixteenth of an inch. 11. Divide a line 5 in. long in extreme and mean ratio. Measure the two segments and determine their lengths to the nearest sixteenth of an inch. 12. Divide a line 6 in. long in extreme and mean ratio. Measure the two segments and determine their lengths to the nearest sixteenth of an inch. The propositions on this page are taken from recent college entrance examination papers. |