1. Convince yourself by measurement and by paper cutting that from every trapezoid there may be cut a triangle (or triangles) which when properly adjusted to another part (or parts) of the trapezoid, will convert the trapezoid into a rectangle. 2. Convince yourself that the rectangle made from a trapezoid is not so long as the longer of the parallel sides of the trapezoid, and not so short as the shorter of the parallel sides of the trapezoid—that its length is midway between the lengths of the two parallel sides of the trapezoid. Note.-Observe that the length of the rectangle thus formed may be found by adding half the difference of the parallel sides of the trapezoid to its shorter side, or by dividing the sum of its parallel sides by 2. 3. To find the area of a trapezoid, find the area of the rectangle to which it is equivalent, or, as the rule is usually given, -"Multiply one half the sum of the parallel sides by the altitude." PROBLEMS. 1. Find the area of a trapezoid whose parallel sides are 10 inches and 15 inches respectively, and whose altitude is 8 inches. 2. How many acres in a trapezoidal piece of land, the parallel sides being 28 rods and 36 rods respectively, and the breadth (altitude) 25 rods? 294. MISCELLANEOUS REVIEW. 1. If 3 men can build 72 feet of sidewalk in a day, how many feet can 4 men build? 2. If 3 men can do a piece of work in 12 hours, in how many hours can 4 men do an equal amount of work? 3. If a piece of land 8 rods square is worth $500, how much is a piece of land 16 rods square worth at the same rate ?* 4. If a ball of yarn 3 inches in diameter is enough for a pair of stockings, how many pairs of stockings can be made from a ball 6 inches in diameter ?+ 5. If a grindstone 12 inches in diameter weighs 40 lb., how much will a grindstone 18 inches in diameter weigh, the thickness and quality of material being the same? 6. The opening in an 8-inch drain tile is how many tiines as large as the opening in a 2-inch drain tile? I 7. Find the area of a rhomboidal piece of land whose length (base) is 64 rods, and whose width (altitude) is 15 rods. 8. Find the area of a trapezoidal piece of land, the length of the parallel sides being 44 rods and 52 rods respectively, and the width (altitude) being 18 rods. § 9. Find the area of a triangular piece of land whose base is 42 rods and whose altitude is 20 rods. * Make diagrams of these pieces of land on a scale of 4 rods to the inch. + Compare a 3-inch cube and a 6-inch cube. Remember that a 3-inch sphere is a little more than half of a 3-inch cube, and a 6-inch sphere a little more than one half of a 6-inch cube. # Compare a 6-inch square with a 2-inch square. Remember that a 2-inch circle is about of a 2-inch square, and an 8-inch circle about of an 8-inch square. $ Draw a diagram of the land on a scale of 1 inch to the rod. POWERS AND ROOTS. 295. A product obtained by using a number twice as a factor is called the second power or the square of the number; thus, 25, (5 x 5), is the second power, or the square of 5. NOTE.—Twenty-five is called the second power of 5, because it may be obtained by using 5, twice as a factor. It is called the square of 5, because it is the number of square units in a square whose side is 5 linear units. 8 ? 1. What is the second power of 2 ? 3 ? 5? 2. What is the square of 4? 7? 1? 6? 9? 10 ? (a) Find the sum of the eighteen squares. 296. The square root of a number is one of the two equal factors of the number. The radical sign, V, (without a figure above it) indicates that the square root of the number following it, is to be taken; thus 64, means the square root of 64. 1. What is the square root of 144 ? 81 ? 49? ✓9 = ? ✓ 64 = ? ✓121 = ? ✓100 = ? ✓169 ? (b) Find the sum of the fourteen results. Powers and Roots. 297. Any number that can be resolved into two equal factors is a perfect square. 1. Tell which of the following are perfect squares and which are not: 9, 10, 12, 16, 18, 25, 32, 36. NOTE. - It is a curious fact that no number, either integral or mixed, can be found which, when multiplied by itself, will give as a product 10, or 12, or 14, or any number that is not a perfect square. 2. Any integral number that is a perfect square is composed of an even number of like prime factors; that is, its prime factors are an even number of 2's, 3's, 5's, 7's, etc. 3. Tell which of the following are perfect squares ? 144, (2 x 2 x 2 x 2 x 3 x 3); 250, (2 x 5 x 5 x 5); 225, (5 x 5 x 3 x 3). RULE. - To find the square root of an integral number, that is a perfect square, resolve the number into its prime factors and take half of them as factors of the root; that is, one half as many 2's, 3's, or 5's, etc., as there are 2's, 3's, or 5's, etc., in the factors of the number. 4. Find the square root of 1225. 1225 5 x 5 x 7 x 7. ✓1225 = 5 x 7 = 35. 5. ' Find the square root of 441; of 400. 6. Find the square root of 576; of 324. 7. Find the square root of 784; of 2025. 8. Find the square root of 625; of 3025. (a) Find the sum of the last eight results. Powers and Roots. 298. THE SQUARE OF COMMON FRACTIONS. 1. The square of 1 (1 x D), is - -. NOTE.-A square whose side is į (of a linear unit) has an area of \ (of a square unit). Show this by diagram. 2. Answer the following and illustrate by diagram if necessary: (3) = ? (1)2 = ? (1)2 = ? (&)2 = ? (x) = ? 2 (a) Find the sum of the eight results. 3. A square of sheet brass whose edge is of a foot is what part of a square foot ? 299. THE SQUARE Root OF COMMON FRACTIONS. 1. The square root of 16 is Note 1.-A square whose area is 16 (of a square unit) is (of a linear unit) in length. Show this by diagram. Note 2.- Only those fractions are perfect squares, which, when in their lowest terms, have perfect squares for numerators and perfect squares for denominators. 2. What is the square root of 6 ? Of is? Of 1? V 14 = ? ? ✓98 = ? Viii = ? (b) Find the sum of the seven results. 3. The area of a square piece of sheet brass is 84 of a square foot. What is the length of the side of the square? 4. How long is the side of a square of zinc the area of which is of a square yard ? 5. What is the perimeter of a square piece of land whose area is jó of a square mile ? 169 |