EXERCISE 150 Extract, to three decimal figures, the square root of: PROBLEMS INVOLVING SQUARE ROOT 1. The area of a square field is 1 A. in yards of one of its sides. 2. The area of a square field is 12 A. in yards of one of its sides. Find the length Find the length 3. The dimensions of a rectangle are 289 yd. and 196 yd. Find the side of an equivalent square. 4. The dimensions of a rectangle are 14 mi. and .7 mi. Find, correct to four decimal figures, the side of an equivalent square. 5. Find in rods the perimeter of a square field whose area is of a square mile. 6. The area of a rectangle whose length is twice its breadth is 10 A. Find its dimensions in yards. HINT. Draw a diagram; divide it into two equal parts by a line parallel to its width. Notice what each part is. 7. The area of its width is 20 A. a rectangle whose length is three times Find its dimensions in yards. 8. A square and a rectangle have the same area; namely, 40 A. If the length of the rectangle is twice its width, find in rods the difference between their perimeters. The side of a right triangle opposite the right angle is called the hypothenuse. The other two sides are called the legs of the right triangle. One of the legs is called the base of the right triangle, and the other leg is called the altitude of the right triangle. In a right triangle the square on the hypothenuse is equal to the sum of the squares on the two legs. Thus in the right triangle of the figure, if a is 6 ft., b is 8 ft., and c is 10 ft., a2 + b2 = c2, or 62+ 82 = 102, which we can see is true; because 36 +64 = 100. This fact gives a method of finding one side of a right triangle if the other two are given. a Example. In a right triangle the legs are 7 and 24. Find the hypothenuse. 1. In a right triangle, given a = 6, b = 8, find c. 9. In a right triangle, given a = 51, b = 140, find c. 10. In a right triangle, given a = 44, b = 52.5, find c. 11. In a right triangle, given a = 87, b = 416, find c. 12. In a right triangle, given a = 136, b = 273, find c. 13. In a right triangle, given a = 145, b = 408, find c. 14. In a right triangle, given a = 207,6 224, find c. = 15. A ladder is placed 14 ft. from a wall 48 ft. high. How long must the ladder be to reach to the top of the wall? 16. Find the length of the diagonal of a square if one side of the square is 10 rods. 17. Find the length of the diagonals of a rectangle, the dimensions of the rectangle being 17 rd. and 25 rd. Example. If the hypothenuse of a right triangle is 493 and one leg is 468, find the other leg. SOLUTION. Let the required leg be a. 1. Hypothenuse 377, base 345, find the altitude. = = = = 2. Hypothenuse = 545, base = 513, find the altitude. 3. Hypothenuse 449, base = 351, find the altitude. 4. Hypothenuse 5.05, base = 4.56, find the altitude. 5. Hypothenuse = .461, base = .38, find the altitude. 6. Hypothenuse = .481, alt. = .36, find the base. 7. Hypothenuse = .641, alt. = .609, find the base. 8. Hypothenuse = .773, alt. .195, find the base. 9. Hypothenuse = .697, alt..528, find the base. AREAS OF PLANE TRIANGLES The following rule gives the area of any triangle: (1) Add the three sides and take half the sum. (2) Subtract each side separately from the half sum. (3) Find the continued product of the three remainders and the half sum. (4) The square root of this product is the area. This rule enables one to find the area of irregular tracts of land which can be divided into triangles whose sides can be measured. Example. Find the area of a triangle whose sides are 34 ch., 65 ch., and 93 ch. 96 Area = √96 × 62 × 31 × 3 = √553535 =744. .. area = 744 sq. ch. 74.4 A. = EXPLANATION. 192 is the sum of the sides. 96 is the half sum of the sides. 62, 31, and 3 are the remainders obtained by subtracting each side from the half sum of the sides. 553,535 is the continued product of the three remainders and the half sum. 744 is the square root of this product. EXERCISE 154 Find the area of each of the following triangles: 1. Given the sides, 13, 20, 21. 2. Given the sides, 13, 30, 37. 3. Given the sides, 33, 34, 65. 8. Given the sides, 46 rd., 75 rd., 109 rd. 14. Find the area of an isosceles right triangle if the hypothenuse is 27 inches. 15. Find the area of a square whose diagonal is 72 feet. 16. Find the side of a square equivalent to the difference of two squares whose sides are 89 feet and 68 feet. MENSURATION OF THE CIRCLE, ETC. Take a string, and find the length of the circumference of a circle. Take another string and find the length of the diameter of the circle. Divide the former result by the latter to get the ratio of the circumference of the circle to its diameter. The ratio of the circumference of a circle to its diameter is approximately 3.1416. This ratio is denoted by the Greek letter π (Pi). In cases where the numbers involved are not very large, or where extreme accuracy is |