Find the largest square in the first period on the left, and write its root as the first figure in the required root. Subtract its square from the period and annex the second period to the remainder. Double the root found for a trial divisor. Divide the remainder, omitting the last figure, by this trial divisor, and annex the quotient to the trial divisor and also to the root. Multiply the complete divisor by the second figure of the root, and subtract the product from the remainder. Double the root already found for another trial divisor, and proceed as above until all the periods have been used. NOTE. When a naught occurs in the root, annex a naught to the trial divisor, bring down another period, and proceed as before. In extracting the square root of a decimal or a whole number and a decimal, point off into periods of two figures each, the whole number toward the left and the decimal toward the right of the decimal point. In extracting the square root of a fraction, extract the square root of the numerator and of the denominator if both terms are perfect squares: or reduce the fraction to a decimal, and extract the root of the decimal. Find the square root to the nearest hundredth of : The hypotenuse of a right triangle is the side opposite the right angle. 1. How many square units are there in the square described upon the hypotenuse? in the square described upon the perpendicular? in the square described upon the base? 2. How do the number of square units described upon the hypotenuse compare with the sum of the square units described upon the other two sides? That this is universally true is shown by the following diagram: Take any right triangle, as 1; lay it off on a piece of cardboard and draw the square on its hypotenuse. Cut this square into the four equal triangles 1, 2, 3, and 4, and the small square 5, as here shown. By changing the position of the triangles 1 and 2 as indicated, we change the first diagram into the second. But the first is the square on the hypotenuse, and the second is the sum of the squares on the other two sides. Since they are equal, the truth of the proposition is evident. The square on the hypotenuse of a right triangle equals the sum of the squares described on the other two sides. HAM, SCHOOL ARITH.-19 34 ft. 90° 60 ft. Written Work 1. Find the hypotenuse of this right triangle. Hypotenuse2 = 342 + 602, or 4756 Draw figures to a convenient scale and find the unknown 7. Find the length of the longest straight line that can be drawn on a table 8 ft. by 4 ft. B has a 8. A has a field 40 rd. long and 30 rd. wide. square field whose side equals the diagonal of A's field. What is the difference in the area of the two fields? 9. Find the longest straight line in a room 16 ft. in length by 12 ft. in width by 10 ft. in height. 10. Two automobiles, A and B, start from the same point. A goes east 10 mi., then north 10 mi.; B goes west 20 mi., then south 20 mi. Draw figure and find distance apart. 11. Find the length of the diagonal of a square field containing 225 sq. rd. 12. Find the side of the largest square that may be inscribed in a circle 2 ft. in diameter. 13. A fireman's ladder just reaches a window 36 ft. above the ground. How long is the ladder if its foot is 27 ft. from the building? MENSURATION Rectangular surfaces, rectangular solids, and the cylinder have been treated under Practical Measurements. REGULAR POLYGONS A plane is a surface such that a straight line joining any two of its points lies wholly in the surface. A polygon is a plane figure bounded by straight lines. A regular polygon is a plane figure having equal sides and equal angles. TRIANGLE SQUARE PENTAGON HEXAGON Polygons are named from their sides. A regular polygon of three sides is called an equilateral triangle; one of four sides, a square; one of five sides, a pentagon; one of six sides, a hexagon, etc. Finding the area of a regular polygon. Draw a circle. Divide the circumference into 6 equal parts by points marked at distances apart equal to the length of the radius. Join these points, thus making a hexagon. Connect the opposite points by dotted lines, thus dividing the hexagon into six equilateral triangles. Show by folding together the opposite sides of any equilateral, and the equal sides of any isosceles triangle, that each may be divided into two equal right triangles. The area of any regular polygon equals the sum of the areas of the triangles composing it. 1. Find the surface of the bottom of a hexagonal silo that is 12 feet on a side, the distance from the middle point of the side to the center of the bottom being 10.3 ft. 2. How far from the corner is the center of a square field that is 40 rods on a side? (Draw the figure.) 3. Find the area of an equilateral triangular design that is 15 inches on a side. (Divide into two right triangles.) SOLIDS A solid is anything that has length, breadth, and thickness. The faces of a solid are the surfaces that bound it. The lateral or convex surface of a solid is the area of its sides, or faces. The volume of a solid is the number of cubic units it contains. A prism is a solid whose ends are equal and parallel polygons, and whose sides are parallelograms. Prisms are named from their bases, as triangular, square, rectangular, pentagonal, hexagonal, etc. TRIANGULAR SQUARE PRISM PENTANGULAR PRISM RECTANGULAR PRISM A cylinder is a solid with circular ends and uniform diameter. The ends are the bases, and the curved surface is the convex surface. The altitude of a prism or of a cylinder is the perpendicular distance between the bases. |