ever are together equal to the square upon the hypothenuse? 241. Construct a triangle, whose base shall be 12, and the sum of the other two sides 15, and of which one side shall be twice the length of the other. 242. Can you make one square that shall be equal to the sum of two other squares? 243. Can you make a square that shall equal the difference between two squares? 244. Can you make a square that shall equal in surface the sum of three squares. The angle made by the two lines joining the centre of a polygon with the extremities of one of its sides is called the angle at the centre of the polygon; and the angle made by any two contiguous sides of a polygon is called the angle of the polygon. 245. Make an octagon in a circle, measure by a line of chords the angle at the centre and the angle of the octagon, and prove the correctness of your work by calculation. A scale having its breadth divided into ten equally long and narrow parallel spaces, cut at equal intervals by lines at right angles to them, with a spare end division subdivided similarly, only at right angles to the other divisions, into ten small rectangles, each of which small rectangles, being provided with a diagonal, is called a diagonal scale. 246. Make a diagonal scale that shall express a number consisting of three digits. 247. With the assistance of a diagonal scale, construct a plan of a rectangular piece of ground, whose length is 556 yards, and breadth 196 yards, and divide it by lines parallel to either end into four equal and similar gardens, and name the area of the whole piece and of each garden. When a pyramid is divided into two parts by a plane parallel to the base, that part next the base is called a frustum of that pyramid. 248. Make of one piece of card the frustum of a pentagonal pyramid, and let the small end of the frustum contain one-half the surface of that which the greater end contains. 249. Out of a piece of paper, having irregular boundaries to begin with, make a square, using no instruments besides the fingers. 250. Can you show by a figure in what cases the square of is of the same value as of, and in what cases the square of is of greater value than of? 251. Construct, by a diagonal scale, a triangle whose three sides shall be equal to 791, 489, and 568. 252. Can you show to the eye how much is greater than ? 253. How many ways can you show of drawing one line parallel to another line, and through a given point? 254. Show by a figure how many square inches there are in a square whose side is 11 inch, and prove the truth of the result by arithmetic. 255. Show by a figure how many square yards there are in a square pole. You know how to find the area of a rectan gle, and you have changed a rectangle into a rhomboid. 256. How would you find the area of a rhombus ? 257. Can you make a right-angled isosceles triangle equal to a square? 258. Can you make a circle half the size of another circle? 259. Can you make an equilateral triangle double the size of another equilateral triangle? 260. Make of one piece of cardboard a hollow rhombic prism; show how you arrange the sides to fit; and give a sketch of the prism when complete. 261. Make a square, whose length and breadth are 6, and make rectangles, whose lengths and breadths are 7 and 5, 8 and 4, 9 and 3, 10 and 2, and 11 and 1, and show that, though the sums of the sides are all equal, the areas are not all equal. 262. What is the largest rectangle that can be placed in an isosceles triangle? 263. Show by a figure which is greater, and how much, 2 solid inches or 2 inches solid. If from one extremity of an arc there be a line drawn at right angles to a radius joining that extremity, and produced until it is intercepted by a prolonged radius passing through the other extremity, such line is called the tangent of that arc. You have given an example of a tangent to a circle. 264. Give an example of a tangent to an arc. 265. Can you draw a tangent to an arc of 90° ? 266. Can you contrive to place on one line the tangents to the arcs of all the degrees, from that of one degree to that of about 85°; i. e., can you make a line of tangents? 267. Show which tangent, or rather, the tangent to which arc, is equal to the radius of the line of tangents. 268. Make, by the line of tangents, angles of 20°, 40°, 75°, and 80°. |