Miscellaneous Exercises Ex. 161. AOB is a diameter of O 0. C is any point of AB. D is the mid-point of BC and E is the mid-point of AC. Prove ▲ DOE is a right angle. Suggestion. - Draw CO. Ex. 162. Points A and B are on the diameter XY of circle O at equal distances from 0. CA and DB are perpendicular to XY, meeting the semicircle at C and D respectively. Prove ABDC is a rectangle. Ex. 163. If a circle is inscribed in a right triangle, the sum of its diameter and the hypotenuse is equal to the sum of the legs of the triangle. B E F Ex. 164. If AB is a common external tangent of two circles which touch each other externally at C, prove ACB is a right angle. Suggestion. Draw the common tangent of the at C, meeting AB at D. Ex. 165. Prove that the bisector of the angle between two tangents to a circle passes through the center of the circle. Suggestions. -Draw radii to the points of contact. Recall § 120. A Ex. 166. The circle drawn on one of the equal sides of an isosceles triangle as diameter bisects the base. Ex. 167. Two circles are tangent externally at C. In one circle ▲ ABC is inscribed, having one vertex at the point of contact of the circles. AC and BC are extended through C, meeting the other circle at D and E respectively. Prove DE || AB. Suggestion. Draw the common tangent through point C. Ex. 168. If a straight line be drawn through the point of contact of two circles which are tangent externally, terminating in their circumferences, the tangents at its extremities are parallel. Suggestion. -Draw the common internal tangent of the circles. Ex. 169. If AB and AC are the tangents from point A to the circle O, Z BAC = 2 Z OBC. Suggestions.-1. Draw OA. What relation does it bear to BC? 2. Compare LBAO with OBC. Ex. 170. Euclid's construction for the tangent to a circle with center M from a point A outside of it is as follows: 1. Draw the circle with center M and radius MA. 2. Draw MA intersecting the given O at B. 3. Draw BC1 MA at B, meeting the larger at C. 4. Draw MC, intersecting the given O at D. Statement. AD is tangent to the given O. Make the construction and give the proof. Ex. 171. Given a side and the diagonals of a parallelogram, construct the parallelogram. Ex. 172. Through a given point within a circle, construct a chord equal to a given chord. Is there any restriction on the location of the point? Ex. 173. Construct a parallel to the side BC of AABC meeting AB and AC at D and E respectively, so that DE will equal EC. Ex. 174. If point B bisects arc AC of a circle, then A of ▲ ABC equals ▲ C. Ex. 175. Prove that the bisectors of the angles of a circumscribed quadrilateral pass through a common point. Ex. 176. Prove that two chords which are perpendicular to a third chord at its extremity are equal. Ex. 177. If A YC and BC A AXRAYCS. Ex. 178. = In the figure for Ex. 177 draw AC cutting XB at M and YB at N. Prove ▲ AXM≈ ▲ YCN. B Ex. 179. A carpenter has a tool called a gauge which illustrates and applies one of the fundamental loci theorems. The shaded rectangle represents the end of a board; the tool is upon the right-hand side of the board. P is a marking point which extends to the under side of the tool. AB is a movable part which can be fixed at any short distance from P by means of a screw at A. By moving the gauge so that AB is constantly against the edge of the さ RA B board, the point P traces upon the upper side of the board a line parallel to the edge of the board. Why is this so? PROPORTION BOOK III SIMILAR POLYGONS 242. The Ratio of one number to another is the quotient of the first divided by the second. Thus, the ratio of a to b is a ; it is also written a: b. b The numerator is called the Antecedent and the denominator is called the Consequent. Since a ratio is a fraction, it is subject to the usual rules for operations with fractions. 243. The ratio of two concrete quantities of the same kind is the ratio of their measures in terms of a common unit. (§ 212.) Thus, the ratio of 350 lb. to 2 tons is or. $50% Ex. 1. Express the following ratios in their simplest form. (a) 3 to 9. (b) 12 to 2. (c) 5x to 2x. (e) to . (g) 25 to 375. (h) a2 - b2 to a3 — b3. line 15 in. long is divided into two parts which have the Find the parts. Ex. 2. A ratio 23. Suggestion. If the short part contains x in. and the long part (15 — x) in., then x 2 Ex. 3. Complete the solution. Divide a line 63 in. long into two parts whose ratio is 3 : 4. Divide 36 into two parts such that the ratio of the greater diminished by 4 to the less increased by 3 will be 3 : 2. Ex. 5. The ratio of the height of a tree to the length of its shadow on the ground is 17 20. Find the height of the tree if the length of the shadow is 110 feet. Ex. 6. What is the ratio of: (a) a right angle to a straight angle? (b) a right angle to the perigon? (c) one angle of an equilateral triangle to the sum of all the angles of the triangle? (d) one side of a square to the perimeter of the square? as, 244. A Proportion is a statement that two ratios are equal; This proportion is read "a is to b as c is to d." Thus, 1, 3, 5, and 15 form a proportion since 1 5 3 15 This means that 1 bears to 3 the same relation that 5 bears to 15. The first and fourth terms of a proportion are called the Extremes, and the second and third terms, the Means. In the proportion ab c:d, a and d are the extremes and b and c are the means; a and c are the antecedents, and b and d are the consequents. Ex. 7. Select four numbers which form a proportion like the arithmetical illustration in § 244. Ex. 9. Find the value of the literal number in each of the following Ex. 10. Find the value of x in each of the following proportions. 245. Proportion is used in a great variety of ways. EXAMPLE. -The cost of a number of articles of a given kind is "proportional" to the number of articles. Thus, the cost of seven books is to the cost of three books of the same kind as 7 is to 3. Hence, if 3 books cost $1.35, the cost of 7 books may be determined from the proportion = х 7 1.35 3 Ex. 11. Determine by proportion the cost of 13 yd. of cloth if the cost of 5 yd. of the same cloth is 70%. Ex. 12. Determine by proportion the distance an automobile will travel in one hour if it travels 2 mi. in 5 minutes. Ex. 13. If a girl makes $14.25 profit from 15 hens in one year, what profit can she expect from 50 hens, assuming the same average profit per hen? 246. The Fourth Proportional to three numbers a, b, and c is the number x in the proportion a: b = c: c: x. Thus, the fourth proportional to 2, 3, and 4 is the number x in .. 2 x = 12, or x = = 6. Note. 2 4 3 The numbers must be placed in the proportion in the order in which they are given as in the illustrative example. 247. The Third Proportional to two numbers a and b is the number x in the proportion a : b = b: x. 248. A Mean Proportional between two numbers a and b is the number x in the proportion a: x = x: b. EXAMPLE. A mean proportional between 2 and 3 is x in: There are two mean proportionals between any two numbers. The positive one is implied when "the" mean proportional is specified. Ex. 16. Find the mean proportional between : (a) 75 and 12. (b) 3a and a. (c) 2 r3t and 18 rt. (d) 63 and 3. |