PROPOSITION XXVII. THEOREM 546. The perimeters of similar polygons are to each other as any two homologous sides. 548. EXERCISE. The perimeters of similar triangles are to each other as any homologous altitudes. 549. EXERCISE. The perimeters of similar triangles are to each other as any homologous medians. 550. EXERCISE. The perimeters of two similar polygons are 78 and 65; a side of the first is 9, find the homologous side of the second. 551. DEFINITION. A line is divided into extreme and mean ratio when it is divided into two parts so that one segment is a mean proportional between the whole line and the other segment. PROPOSITION XXVIII. PROBLEM 552. To divide a line into extreme and mean ratio. E A Let AB be the given line. Required to divide AB into extreme and mean ratio. Draw BC to AB and equal to one half of AB. Draw AC. With C as a center and CB as a radius describe a circle cutting AC at D, and AC prolonged at E. Lay off AF = AD. 553. EXERCISE. To determine the values of the segments of a line that has been divided into extreme and mean ratio. In the figure of § 552, let the length of AB be a; AF = x, then FB ax. Substituting these values in the last proportion, we get 554. EXERCISE. Divide a line 5 in. long into extreme and mean ratio, and calculate the value of the segments, PROPOSITION XXIX. PROBLEM 555. To draw a common tangent to two given circles. Let A and B be the centers of the two given circles. Divide AB (internally and externally) at c so Draw CD tangent to circle B. Draw the radius BD. [It is required to show that AE = R.] ▲ AEC and CBD are similar (?), whence AC AE = BC BD R AE and AE = R, and ED is a common tangent. r = r 556. DEFINITION. Q.E.F. The two tangents that pass through the internal point of division of AB are called the transverse tangents. The two tangents that pass through the external point of division are called the direct tangents. The points of division are called the centers of similitude of the two circles. 557. EXERCISE. The line joining the centers of two circles is divided harmonically by the centers of similitude. 558. EXERCISE. The line joining the extremities of parallel radii of two circles passes through their external center of similitude if the radii are turned in the same direction; but through their internal center if they are turned in opposite directions. 559. EXERCISE. All lines passing through a center of similitude of two circles and intersecting the circles are divided by the circumferences in the same ratio. Similarly, show that D, E', and F' are in a straight line, also E, D', and F', and also F, D', and E'. 7. The shadow cast by a church steeple on level ground is 27 yd., while that cast by a 5-ft. vertical rod is 3 ft. long. How high is the steeple? 8. The line joining the middle points of the non-parallel sides of a trapezoid circumscribed about a circle is equal to one fourth the perimeter of the trapezoid. [See § 396.] B A D |