169. How can a tape line be used to place four stakes so that they will determine two parallel lines?

170. In the left-hand figure below, the distance between A and B, two points on opposite sides of a river, is desired but cannot be measured directly. By using a tape what lines in the neighborhood of A can be laid out and measured so that the distance AB can then be computed? Assume values for the measured lines and compute AB.

171. In the right-hand figure above, the points A and B are accessible, but the line AB cannot be directly measured. What lines can be laid out and measured that will enable one to compute AB?` Assume values for the measured lines and compute AB.

172. In surveying practice a line is prolonged through an obstacle a given distance by the following method: In the adjacent figure suppose it is required to prolong KB through O so

making an acute angle with AB. Locate C and D so that AE= 2 AC

2 BC. Lay out FDGH so 2 FG. Lay out EGI so that EI

= 4 CD. Lay out BCF so that BF: that FH 4 FD 2 EG. Then IH will satisfy the conditions set. Prove that this is true. The

surveyor checks the accuracy of his work by measuring IH. If it equals AB, he considers his work is correct. Prove that this checks the work.

173. In the adjacent figure the distance AB between two inaccessible points offshore is desired. How must the points K, R, L, M, and H (and other points if necessary) be located, and what lines must be measured in order to com

K

H

KR

B

L

M

pute AB? Assume numerical values for the measured lines and compute AB..

MISCELLANEOUS NUMERICAL EXERCISES

174. Two straight stretches of railroad track, IA and HB, are connected by a circular curve. The chord AB is 100 feet. The distance from the center of the chord to the nearest point of the

track is 8 feet. Find the radius of the curve of the inner rail.

A

175. In triangle ABC, b is 8, a is 12, and c is 10. Find the segments of a made by the bisector of ZA.

B

H

176. In the triangle of Ex. 175 find the segments of a made by the bisector of the exterior angle at A.

177. CK is the altitude on the hypotenuse AB of the triangle ABC. AB is 25 and CK is 12. Find AK and AC.

178. In the triangle of Ex. 177, if AB is 13 and AK is 4, find AC and CK.

179. In the triangle ABC, c is 18, b is 20, and a is 24. Find the altitude on side 18.

180. For the triangle of Ex. 179 find the diameter of the circumscribed circle.

181. The altitude of an equilateral triangle is 8. Find to three decimals the length of one side.

182. ABC and AKR are secants to the same circle. AB is 12, chord BC is 8, and chord KR is 15. Find AK and the tangent to the circle from A.

183. The radii of two circles are 8 and 20 respectively. The distance between their centers is 40. How far from the center of the smaller circle does (a) the external common tangent cut the line of centers? (b) the internal common tangent?

184. AB and KR are two chords of a circle intersecting at O. AB is 24, OR is 18, and OK is 3. Find AO and OB.

185. The hypotenuse of a right triangle is 37 and another side is 35. The perimeter of a similar triangle is 308. Find the shortest side of the second triangle.

186. In the triangle ABC, c is 8 inches, A is 45°, and B is 60°. Find b and a.

187. In the triangle ABC, c is 10, b is 16, and ZA is 30°. Find a.

188. How far is a swimmer from a cliff 400 feet high if when his eye is level with the surface of the water the top of the cliff is just visible? (Use 3960 miles for the earth's radius.)

HINT. Let h = the height of the mountain and d the distance required, both in miles. Then d2 = h (h + 2R).

189. How high is a mountain if a swimmer thirty miles away can see its top from the surface of the ocean?

HINT. h (h+2R) = d2. Since his small compared to 2R, this becomes very closely 2 Rh = d2. Whence h =

d2 2 R

190. If h is one's height in feet above the surface of the ocean and d the distance in miles to an object on its surface just visible

on the edge of the horizon, show that d:

=

3 h
2

approximately.

191. A lookout 100 feet above the surface of the ocean observes an object close to the surface as it comes into view on the edge of the horizon. How far away is the object?

192. K is any point on the semicircle AB. AK cuts the tangent at B in R and BK cuts the tangent at A in L. Prove that AB is a mean proportional between AL and BR.

193. From the vertex D of the parallelogram ABCD a straight line is drawn cutting AB at R and CB produced at K. Prove that CK is a fourth proportional to AR, AD, and AB.

194. AB is a diameter of a circle and KB is a tangent. AK cuts the circle in R. Prove that AR · AK = AB2.

195. Two chords of a circle, AB and CD, are produced to meet in K. Then KR is drawn parallel to AD, meeting CB produced at R. Prove that KR is a mean proportional between BR and CR.

196. If two lines parallel respectively to two sides of a triangle are drawn through the point of intersection of the medians, they trisect the third side.

197. Prove Theorem 20 thus: Draw a line through B and L cutting K'R' at H. Then show by the use of the theorems on proportion that L' coincides with H; that is, that the line through L and L' passes through B.

198. Through K, any point in the common base of the triangles ABC and ABR, lines are drawn parallel to AC and AR, meeting BC and BR in the points F and G respectively. Prove that FG is parallel to CR.

199. Two nonintersecting circles are cut by a third circle. The two common chords intersect at L. Secant

B

C

K G

R

H

LKR cuts the first circle in K and R, and secant LGH cuts the second in G and H. Prove that LK × LR :

= LG X LH.

HINT. Draw the tangents LA, LB, and LC.

Required to construct a line x, so that a: x = x: b.

Construction. Draw the indefinite straight line KG and lay off KL equal to a and LR equal to b. Bisect KR at 0. With O as a center and OK as a radius, construct semicircle KR. Construct a perpendicular to KR at L, cutting the semicircle at II. IIL is the required line.

HINTS. Draw KH and HR and use § 282.

EXERCISES

200. Construct if r = √ab, a and b being lines of given length. 201. Construct a line √6 inches long.

HINTS. Let x be the line. Then x2 = 6. Therefore 2: x = x: 3.

202. Construct a line a √3 inches long, a being a line of given length.

HINTS. Let x be a√3. Then x2

3 a2 = a a (3a). a: x = x : 3 a, etc. 203. Construct x if x is a √, a being a given line.

204. Construct æ if a is√3, a being a given line.