244. A line is drawn parallel to a side AB of a triangle ABC, and cutting AC in D, BC in E. If AD: DC=2: 3, and AB = 20 inches, find DE. 245. The sides of a triangle are 9, 12, 15. Find the segments made by bisecting the angles. (8 313.) 246. A tree casts a shadow 90 feet long, when a vertical rod 6 feet high casts a shadow 4 feet long. How high is the tree? 247. The bases of a trapezoid are represented by a, b, and the altitude by h. Find the altitudes of the two triangles formed by producing the legs till they meet. 248. The sides of a triangle are 6, 7, 8. In a similar triangle the side homologous to 8 is equal to 40. Find the other two sides. 249. The perimeters of two similar polygons are 200 feet and 300 feet. If a side of the first polygon is 24 feet, find the homologous side of the second polygon. 250. How long must a ladder be to reach a window 24 feet high, if the lower end of the ladder is 10 feet from the side of the house? 251. If the side of an equilateral triangle = a, find the altitude. 252. If the altitude of an equilateral triangle = h, find the side. 253. Find the lengths of the longest and the shortest chord that can be drawn through a point 6 inches from the centre of a circle whose radius is equal to 10 inches. 254. The distance from the centre of a circle to a chord 10 inches long is 12 inches. Find the distance from the centre to a chord 24 inches long. 255. 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. 256. The radius of a circle is 6 inches. Through a point 10 inches from the centre tangents are drawn. Find the lengths of the tangents, and also of the chord joining the points of contact. 257. If a chord 8 inches long is 3 inches distant from the centre of the circle, find the radius and the distances from the end of the chord to the ends of the diameter which bisects the chord. from the centre any chord is drawn. What is the product of the ments of the chord? What is the length of the shortest chord be drawn through the point? 259. From the end of a tangent 20 inches long a secant through the centre of the circle. If the exterior segment of t is 8 inches, find the radius of the circle. 260. The radius of a circle is 9 inches; the length of a tan inches. Find the length of a secant drawn from the extrem tangent to the centre of the circle. 261. The radii of two circles are 8 inches and 3 inches, a tance between their centres is 15 inches. Find the lengths of mon tangents. 262. Find the segments of a line 10 inches long divided and mean ratio. 263. The sides of a triangle are 4, 5, 6. Is the largest a right, or obtuse? PROBLEMS. 264. To divide one side of a given triangle into segments p to the adjacent sides. (8 313.) 265. To produce a line AB to a point C'so that AB: AC 266. To find in one side of a given triangle a point whos from the other sides shall be to each other in a given ratio. 267. Given an obtuse triangle; to draw a line from the v obtuse angle to the opposite side which shall be a mean p between the segments of that side. 268. Through a given point P within a given circle to dra AB so that AP: BP-2: 3. 269. To draw through a given point P in the arc subtended AB a chord which shall be bisected by AB. 270. To draw through a point P, exterior to a given ciro PAB so that PA: AB-4: 3. 271. To draw through a point P, exterior to a given circ PAB so that AB2 PAX PB. = 272. To find a point P in the arc subtended by a given ch that PA: PB = 3 : 1, 273. To draw through one of the points of intersection of two circles a secant so that the two chords that are formed shall be to each other in the ratio of 3: 5. 274. To divide a line into three parts proportional to 2, 4, 4. 275. Having given the greater segment of a line divided in extreme and mean ratio, to construct the line. 276. To construct a circle which shall pass through two given points and touch a given straight line. 277. To construct a circle which shall pass through a given point and touch two given straight lines. 278. To inscribe a square in a semicircle. 279. To inscribe a square in a given triangle. M HINT. Suppose the problem solved, and DEFG the inscribed square. Draw CM || to AB, and let AF produced meet CM in M. Draw CH and MN 1 to AB, and produce AB to meet MN at N. The ▲ ACM, AGF are similar; also the A AMN, AFE are similar. By these triangles show that the figure CMNH is a square. By constructing this square, the point F can be found. ADHE B 280. To inscribe in a given triangle a rectangle similar to a given rectangle. 281. To inscribe in a circle a triangle similar to a given triangle. 282. To inscribe in a given semicircle a rectangle similar to a given rectangle. 283. To circumscribe about a circle a triangle similar to a given triangle. 2 abc that is, 2 284. To construct the expression, x= 285. To construct two straight lines, having given their sum and their ratio. 286. To construct two straight lines, having given their difference and their ratio. 287. Having 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 circumferences at B and C, so that AB: AC=1: 2. HINT. Suppose the problem solved, join OA and produce it to D, making QA AD 1.2 Join DC. A OAR ADC are similar BOOK IV. AREAS OF POLYGONS. 358. The area of a surface is the numerical measur surface referred to the unit of surface. The unit of surface is a square whose side is a unit as the square inch, the square foot, etc. 359. Equivalent figures are figures having equal a PROPOSITION I. THEOREM. 360. The areas of two rectangles having equ tudes are to each other as their bases. Let the two rectangles be AC and AF, hav same altitude AD. Proof. CASE I. When AB and AE are commensu Suppose AB and AE have a common measure which is contained in AB seven times and in AE for Apply this measure to AB and AE, and at th points of division erect Is. The rect. AC will be divided into seven rectangle and the rect. AF will be divided into four rectangles CASE II. When AB and AE are incommensurable. D C D H Divide AB into any number of equal parts, and apply one of them to AE as often as it will be contained in AE. Since AB and AE are incommensurable, a certain number of these parts will extend from A to a point K, leaving a remainder KE less than one of the parts. Draw KH to EF. Since AB and AK are commensurable, rect. AH AK rect. AC AB Case I. These ratios continue equal, as the unit of measure is indefinitely diminished, and approach indefinitely the limiting ratios rect. AF AE and rect. AC. AB respectively. limits are equal). $ 250 (if two variables are constantly equal, and each approaches a limit, the Q. E. D. 361. COR. The areas of two rectangles having equal bases are to each other as their altitudes. For AB and AE may be considered as the altitudes, AD and AD as the bases. NOTE. In propositions relating to areas, the words "rectangle," "triangle," etc., are often used for area of rectangle," angle," etc. area of tri |