PROBLEMS OF CONSTRUCTION. Ex. 290. To divide one side of a given triangle into segments propor tional to the adjacent sides (§ 348). B B F B Ex. 291. To find in one side of a given triangle a point whose distances from the other sides shall be to each other in the given ratio m: n. Take AG = m 1 to AC, GH = n 1 to BC. Draw CD to OG. Ex. 292. Given an obtuse triangle; to draw a line from the vertex of the obtuse angle to the opposite side which shall be the mean proportional between the segments of that side. Ex. 293. Through a given point P within a given circle to draw a chord AB so that the ratio AP: BP shall equal the given ratio m: n. Draw OPC so that OP: PCn: m. Draw CA equal to the fourth proportional to n, m, and the radius of the circle. Ex. 294. To draw through a given point P in the arc subtended by a chord AB a chord which shall be bisected by AB. On radius OP take CD equal to CP. Draw DE || to BA. Ex. 295. To draw through a given external point P a secant PAB to a given circle so that the ratio PA: AB shall equal the given ratio m : n. Ex. 296. To draw through a given external point P a secant PAB to a given circle so that AB2 PA x PB. = Ex. 297. To find a point P in the arc subtended by a given chord AB so that the ratio PA : PB shall equal the given ratio m: n. Ex. 298. To draw through one of the points of intersection of two circles a secant so that the two chords that are formed shall be in the given ratio m: n. Ex. 299. Having given the greater segment of a line divided in extreme and mean ratio, to construct the line. Ex. 300. To construct a circle which shall pass through two given points and touch a given straight line. Ex. 301. To construct a circle which shall pass through a given point and touch two given straight lines. Ex. 302. To inscribe a square in a semicircle. Draw CM to AB, meet Ex. 303. To inscribe a square in a given triangle. Let DEFG be the required inscribed square. ing AF produced in M. Draw CH and MNL to AB, and produce AB to meet MN at N. The ACM, AGF are similar; also, the AAMN, AFE are similar. triangles show that the figure CMNH is a square. constructing this square, the point F can be found. By these Ex. 304. To inscribe in a given triangle a rectangle similar to a given rectangle. Ex. 305. To inscribe in a circle a triangle similar to a given triangle. Ex. 306. To inscribe in a given semicircle a rectangle similar to a given rectangle. Ex. 307. To circumscribe about a circle a triangle similar to a given triangle. 2 abc 2 ab с de ; that is, d e Ex. 308. To construct the expression, x = Ex. 309. To construct two straight lines, having given their sum and their ratio. Ex. 310. To construct two straight lines, having given their difference and their ratio. Ex. 311. Given two circles, with centres O and O', and a point A in their plane, to draw through the point A a straight line, meeting the cir cumferences at B and C, so that AB: AC = m : n. PROBLEMS OF COMPUTATION. Ex. 312. To compute the altitudes of a triangle in terms of its sides. B C A D At least one of the angles A or B is acute. Suppose B is acute. Ex. 313. To compute the medians of a triangle in terms of its sides. Ex. 314. To compute the bisectors of a triangle in terms of the sides. Ex. 315. To compute the radius of the circle circumscribed about a triangle in terms of the sides of the triangle. Ex. 316. If the sides of a triangle are 3, 4, and 5, is the angle opposite 5 right, acute, or obtuse? Ex. 317. If the sides of a triangle are 7, 9, and 12, is the angle opposite 12 right, acute, or obtuse? Ex. 318. If the sides of a triangle are 7, 9, and 11, is the angle opposite 11 right, acute, or obtuse? Ex. 319. The legs of a right triangle are 8 inches and 12 inches; find the lengths of the projections of these legs upon the hypotenuse, and the distance of the vertex of the right angle from the hypotenuse. Ex. 320. If the sides of a triangle are 6 inches, 9 inches, and 12 inches, find the lengths (1) of the altitudes; (2) of the medians; (3) of the bisec tors; (4) of the radius of the circumscribed circle. Ex. 321. A line is drawn parallel to a side AB of a triangle ABC, cutting AC in D, BC in E. If AD: DC = 2:3, and AB = 20 inches, find DE. Ex. 322. The sides of a triangle are 9, 12, 15. the sides made by bisecting the angles. Find the segments of Ex. 323. A tree casts a shadow 90 feet long, when a post 6 feet high casts a shadow 4 feet long. How high is the tree? Ex. 324. The lower and upper bases of a trapezoid are a, b, respectively; and the altitude is h. Find the altitudes of the two triangles formed by producing the legs until they meet. Ex. 325. The sides of a triangle are 6, 7, 8, respectively. In a similar triangle the side homologous to 8 is 40. Find the other two sides. Ex. 326. The perimeters of two similar polygons are 200 feet and 300 feet. If a side of the first is 24 feet, find the homologous side of the second. Ex. 327. How long a ladder is required to reach a window 24 feet high, if the lower end of the ladder is 10 feet from the side of the house? Ex. 328. If the side of an equilateral triangle is a, find the altitude. Ex. 329. If the altitude of an equilateral triangle is h, find the side. Ex. 330. Find the length of the longest chord and of the shortest chord that can be drawn through a point 6 inches from the centre of a circle whose radius is 10 inches. Ex. 331. The distance from the centre of a circle to a chord 10 feet long is 12 feet. Find the distance from the centre to a chord 24 feet long. Ex. 332. The radius of a circle is 5 inches. Through a point 3 inches from the centre a diameter is drawn, and also a chord perpendicular to the diameter. Find the length of this chord, and the distance from one end of the chord to the ends of the diameter. Ex. 333. The radius of a circle is 6 inches. Find the lengths of the tangents drawn from a point 10 inches from the centre, and also the length of the chord joining the points of contact. Ex. 334. The sides of a triangle are 407 feet, 368 feet, and 351 feet Find the three bisectors and the three altitudes. |