PROPOSITION XXXII. THEOREM 232. The opposite sides of a parallelogram are equal. To prove AB = CD and BC= AD. The proof is left as an exercise for the student. 233. Cor. I. All the sides of a square are equal, and all the sides of a rhombus are equal. 234. Cor. II. Parallel lines intercepted between the same parallel lines are equal. Cor. III. The perpendiculars drawn to one of two parallel lines from any two points in the other are equal. 236. Cor. IV. A diagonal of a parallelogram divides it into two equal triangles. Ex. 262. The perpendiculars drawn to a diagonal of a parallelogram from the opposite vertices are equal. Ex. 263. The diagonals of a rhombus are perpendicular to each other and so are the diagonals of a square. Ex. 264. The diagonals of a rectangle are equal. Ex. 265. The diagonals of a rhomboid are unequal. Ex. 266. If the diagonals of a parallelogram are equal, the figure is a rectangle. Ex. 267. If the diagonals of a parallelogram are not equal, the figure is a rhomboid. Ex. 268. Draw a line parallel to the base of a triangle so that the portion intercepted between the sides may be equal to a given line. Ex. 269. Explain the statement: Parallel lines are everywhere equidistant. Has this been proved? Ex. 270. Find the locus of a point that is equidistant from two given parallel lines. Ex. 271. Find the locus of a point: (a) one inch above a given horizontal line; (b) two inches below the given line. Ex. 272. Find the locus of a point: (a) one inch to the right of a given vertical line; (b) one inch to the left of the given line. Ex. 273. Given a horizontal line OX and a line OY perpendicular to OX. Find the locus of a point three inches above OX and two inches to the right of OY. 237. Historical Note. René Descartes (1596-1650) was the first to observe the importance of the fact that the position of a point in a plane is determined if its distances, say x and y, from two fixed lines in the plane, perpendicular to each other, are known. He showed that geometric figures can be represented by algebraic equations, and developed the subject of analytic geometry, which is known by his name as Cartesian geometry. Descartes was born near Tours in France, and was sent at eight years of age to the famous Jesuit school at La Flêche. He was of good family, and since, at that time, most men of position entered either the church or the army, he chose the latter, and joined the army of the Prince of Orange. One day, while walking in a street in a Holland town, he saw a placard which challenged every one who read it to solve a certain geometric problem. Descartes solved it with little difficulty and is said to have realized then that he had no taste for military life. He soon resigned his commission and spent five years in travel and study. After this he lived a short time in Paris, but soon retired to Holland, where he lived for twenty years, devoting his time to mathematics, philosophy, astronomy, and physics. His work in philosophy was of such importance as to give him the name of the Father of Modern Philosophy. PROPOSITION XXXIII. THEOREM 238. The diagonals of a parallelogram bisect each other. B C D Given ABCD with its diagonals AC and BD intersecting at 0. To prove 40 = OC and BO= OD. HINT. Prove A OBC = ▲ ODA. .. AO OC and BO OD. = = Ex. 274. If through the vertices of a triangle lines are drawn parallel to the opposite sides of the triangle, the lines which join the vertices of the triangle thus formed to the opposite vertices of the given triangle are bisected by the sides of the given triangle. Ex. 275. A line terminated by the sides of a parallelogram and passing through the point of intersection of its diagonals is bisected at that point. Ex. 276. How many parallelograms can be constructed having a given base and altitude? What is the locus of the point of intersection of the diagonals of all these parallelograms ? Ex. 277. If the diagonals of a parallelogram are perpendicular to each other, the figure is a rhombus or a square. Ex. 278. If the diagonals of a parallelogram bisect the angles of the parallelogram, the figure is a rhombus or a square. Ex. 279. Find on one side of a triangle the point from which straight lines drawn parallel to the other two sides, and terminated by those sides, are equal. (See § 232.) Ex. 280. Find the locus of a point at a given distance from a given finite line AB. Ex. 281. Find the locus of a point at a given distance from a given line and also equidistant from the ends of another given line. Ex. 282. Construct a parallelogram, given a side, a diagonal, and the altitude upon the given side. 239. If the opposite sides of a quadrilateral are equal, the figure is a parallelogram. B A Given quadrilateral ABCD, with AB CD, and BC= AD. Ex. 283. Construct a parallelogram, given two adjacent sides and the included angle. Ex. 284. Ex. 285. Construct a rectangle, given the base and the altitude. Ex. 286. Through a given point construct a parallel to a given line by means of Prop. XXXIV. Ex. 287. Construct a median of a triangle by means of a parallelogram, (1) using §§ 239 and 238; (2) using §§ 220 and 238. Ex. 288. An angle of a triangle is right, acute, or obtuse according as the median drawn from its vertex is equal to, greater than, or less than half the side it bisects. PROPOSITION XXXV. THEOREM 240. If two opposite sides of a quadrilateral are equal and parallel, the figure is a parallelogram. B D Given quadrilateral ABCD, with BC both equal and I to AD. To prove ABCD a . Ex. 289. If the mid-points of two opposite sides of a parallelogram are joined to a pair of opposite vertices, a parallelogram will be formed. Ex. 290. Construct a parallelogram, having given a base, an adjacent angle, and the altitude, making your construction depend upon § 240. Ex. 291. If the perpendiculars to a line from any two points in another line are equal, then the lines are parallel. Ex. 292. If two parallelograms have two vertices and a diagonal in common, the lines joining the other four vertices form a parallelogram. |