139. Construction. Upon a given line-segment corresponding to a side of a given polygon, construct a polygon similar to the given polygon. Required to construct on MN as a side corresponding to AB, a polygon similar to ABCD .... Construction. 1. Draw all possible diagonals from A. 2. At M construct ≤1 = ≤ x, ≤ 2 = 2 y, etc. 3. At N construct 24w, giving point 0. EXERCISES 1. The perimeters of two similar polygons are 144 yd. and 256 yd., respectively. A side of the first is 18 yd. Find the corresponding side of the second. 2. The sides of a polygon are 8 in., 12 in., 15 in., 6 in., and 20 in. The side of a similar polygon corresponding to the 8-in. side is 6 in. Find the perimeter of the similar polygon. 3. The perimeter of a polygon is p and one side is x. If the perimeter of a similar polygon is q, find the side corresponding to x. 4. A rectangular field is w yd. wide and 1 yd. long. Find the perimeter of a similar field 3 w yd. wide. 5. Prove that two corresponding diagonals of two similar polygons have the same ratio as a pair of corresponding sides. 6. Prove that if two polygons are each similar to a third polygon, they are similar to each other. 8. Polygon ABCDE is formed by joining points of intersection of lines on squared paper. Polygon MNOPQ is formed by joining corresponding intersections of lines separated by only one half the interval. Prove that ABCDE~ MNOPQ. 9. Explain how squared paper may be used to construct similar polygons the ratio of whose sides is 3; the ratio of whose sides is 5. 10. Draw any quadrilateral. Upon a given line-segment as a side corresponding to a given side of the quadrilateral, construct by use of compasses and straightedge a similar quadrilateral. 140. Maps and plans. -A map or plan is a figure similar to the figure formed by the object which it represents. Thus, a map of a state is a drawing similar to the figure formed by the state itself. The drawing in the margin shows an architect's floor plan of a house. A map or plan is always drawn to scale, i.e., in the map or plan the distances are made proportional to the actual distances which they represent. Laundry Pantry Kitchen Dining Porch Living Room Thus, a map of the United States which is drawn so that 200 miles measured anywhere across the country is represented on the map by a distance of 1 inch, is said to be drawn to the scale of 200 miles to an inch (scale: 200 mi. = 1 in.). EXERCISES 1. Consult a map of the United States. Find the scale to which it is drawn. Find from this map the number of miles in a straight line from Boston to San Francisco. 2. A map of Illinois drawn to the scale of 200 mi. to an inch is 11⁄2 in. long. How many miles long is the state? 3. On a map drawn to the scale of 240 mi. to an inch, the distance from Chicago to Denver is 3 in. How many miles is it from Chicago to Denver? 4. In the house of which the floor plan is shown in § 140, the width of the living room is 15 ft. By measuring the width of the living room which the plan is drawn. in the plan, find the scale to 5. From the scale found in Ex. 4, determine the number of feet in the width of the porch. Find the length of the porch. 6. How many feet wide is the dining room of this house? 7. Draw a rectangle representing a rectangular field that is 1200 ft. long and 480 ft. wide to a scale of 240 ft. to an inch. What are the dimensions of the drawing? 8. The distance from A to the inaccessible point B may be obtained as follows: Measure a base line AC. Measure ACB and BAC. Then construct a map of the measurements to scale, and determine from the map the distance from A to B. If AC 960 ft., ACB = 40°, and ▲ BAC = = C B 90°, draw a map of the measurements to the scale of 160 ft. to an inch, and compute AB from the map. 9. In order to find the height of a church spire CD, the base line AB is measured 75 ft. long toward the foot of the spire D. It is found that DAC = 50° and DBC =80°. Make a drawing of these measurements to the scale of 25 ft. to an inch, and compute the height CD of the spire. A 10. A and B are two forts in the lines of the enemy, and it is desired to know their distance apart and their distances A from our lines. From point C in our lines, we measure ACD and ▲ BCD. Then we go to a second point D and measure ▲ CDA and ▲ CDB. LACD 120°, LBCD = 50°, = LCDB = 100°, and CD = 2000 ft. to the scale of 500 ft. to the inch, distances AC, BD, and AB. 2 CDA = 45°, B Draw a plan and find the D B In using the plane table for finding the distance between two points M and N, the instrument is set up at any convenient point A. A sheet of paper is fastened on the board, and a pin stuck through the paper into the board at a point a directly over A. The straightedge is placed against the pin, and a base line ab drawn on the paper toward a second point B. Lines an and am are then drawn toward N and M, respectively. The plane table is then removed and set up over point B, so that the line ab on the paper is directly above B and points directly back to the old station A. The M pin is then removed to a point b directly over B, in line ab. Lines bn and bm are then drawn toward N and M, respectively. The distance AB is measured. Lines an and bn meet at n, and am and bm meet at m. measured. Also ab is measured. Then mn is drawn and From these measurements the distance MN is computed by proportion. In using the plane table for making a map or for other kinds of measurements, the procedure is very similar to that described above. EXERCISES 1. Prove that the line MN in the figure of § 141 may be found from |