1 2. The hypothenuse of a right-angled triangle is 53 yd. and the base 84 feet. Find the perpendicular. RULE. Extract the square root of the difference between the square of the hypothenuse and the square of the giver side ; and the result is the required side. 3. A line reaching from the top of a precipice 120 ft. high, on the bank of a river, to the opposite side is 380 ft. long. How wide is the river? Ans. 360 ft. 6 in. +. 4. A ladder 52 feet long stands against the side of a building. How many feet must it be drawn out at the bottom that the top may be lowered 4 feet? Ans. 20 ft. QUADRILATERALS. 735. A Quadrilateral is a plane figure bounded by four straight lines, and having four angles. There are three kinds of quadrilaterals, the Parallelogram, Trapezoid. and Trapezium. . 736. A Parallelogram is a quadrilateral which has its opposite sides parallel. There are four kinds of parallelograms, the Square, Rectangle, Rhom boid, and Rhombus. 737. A Rectangle is any parallelogram having its angles right angles. 738. A Square is a rectangle whose sides are equal. 739. A Rhomboid is a parallelogram whose opposite sides only are equal, but whose angles are not right angles. 740. A Rhombus is a parallelogram whose sides are all equal, but whose angles are not right angles. Square. Rectangle. Rhomboid. Rhombus. 741. A Trapezoid is a quadrilateral, two of whose sides are parallel and two oblique. 742. A Trapezium is a quadrilateral having no two sides parallel. 743. The Altitude of a parallelogram or trapezoid is the perpendicular distance between its parallel sides. The vertical dotted lines in the figures represent the altitude. 744. A Diagonal of a plane figure is a straight line joining the vertices of two angles not adjacent. 745. To find the area of any parallelogram. 1. Find the area of a parallelogram whose base is 16.25 ft. and altitude 7.5 feet. OPERATION.-16.25 x 7.5=121.875 sq. ft., area. 2. The base of a rhombus is 10 ft. 6 in., and its altitude 8 ft. What is its area ? RULE. Multiply the base by the altitude. 3. How many acres in a piece of land in the form of a rhomboid, the base being 8.75 ch. and altitude 6 ch. ? Ans. 51 A. 746. To find the area of a trapezoid. 1. Find the area of a trapezoid whose parallel sides are 23 and 11 feet, and the altitude 9 feet. OPERATION.—23 ft. +11 ft. +2 = 17 ft. ; 17 ft. x 9 = 153 sq. ft., 2. Required the area of a trapezoid whose parallel sides are 178 and 146 feet, and the altitude 69 feet. Ans. 11178 sq. Rule. Multiply one-half the sum of the parallel sides by the altitude. area. ft. 3. How many square feet in a board 16 ft. long, 18 in, wide at one end and 25 in, wide at the other end ? 4. One side of a quadrilateral field measures 38 rd.; the side opposite and parallel to it measures 26 rd., and the distance between the two sides is 10 rd. Find the area. Ans. 2 A. 747. To find the area of a trapezium. 1. Find the area of a trapezium whose diagonal is 42 ft. and perpendiculars to this diagonal, as in the diagram, are 16 ft. and 18 ft. 18 ft. OPERATION.-(18 ft.+ 16 ft. - 2) × 42 = 714 sq. ft., area. 2. Find the area of a trapezium whose diagonal is 35 ft. 6 in., and the perpendiculars to this diagonal 9 ft. and 12 ft. 6 in. RULE. Multiply the diagonal by half the sum of the perpendiculars drawn to it from the vertices of the opposite angles. 3. How many acres in a quadrilateral field whose diagonal is 80 rd. and the perpendiculars to this diagonal 20.453 and 50.832 rd.? To find the area of any regular polygon, multiply its perimeter, or the sum of its sides, by the perpendicular falling from its centre to one of its sides. To find the area of an irregular polygon, divide the figure into triangles and trapeziums, and find the area of each separately. The sum of these areas will be the area of the whole polygon. 748. A Circle is a plane figure bounded by a curved line, called the circumference, every point of which is equally distant from a point within called the center. 749. The Diameter of a circle is a line passing through its center, and terminated at both ends by the circumference. 750. The Radius of a circle is a line extending from its center to any point in the circumference. It is one-half the diameter. PROBLEMS. 751. When either the diameter or the circumference of a circle is given, to find the other dimension of it. 1. Find the circumference of a circle whose diameter is 20 in. OPERATION.—20 in. x 3.1416 = 62.832 in. = 5 ft. 2.832 in., circum. 2. What is the circumference of a wheel 5 ft. 6 in. in diameter. 3. Find the diameter of a circle whose circumference is 62.832 ft. OPERATION.—62.832 ft. +3.1416 = 20 ft., diameter. 4. Find the diameter of a wheel whose circumference is 50 ft. RULE 1. Multiply the diameter by 3.1416; the product is the circumference. 2. Divide the circumference by 3.1416; the quotient is the diameter. 5. What is the diameter of a tree whose girt is 18 ft. 6 in. ? 6. Find the length of tire that will band a wheel 7 ft. 9 in. in diameter. Ans. 24 ft. 4 in. + 7. The diameter of a cylinder is 8 ft. 6 in. ; find its girt. 8. What is the radius of a circle whose circumference is 31.416 ft.? 9. The radius of a circle is 10 ft. ; what is its circumference? 10. Find the circumference of the greatest circle that can be drawn with a string 14 in. long, used as a radius. 7 ft. 3.96 in. 752. To find the area of a circle, when both its diameter and circumference are given, or when either is given. 1. Find the area of a circle whose diameter is 10 ft. and circumference 31.416 feet. OPERATION.-31.416 ft. x 10:4=78.54 sq. ft., area. 2. Find the area of a circle whose diameter is 10 ft. OPERATION.–10 ft.x.7854 = 78.54 sq. ft., area. 3. Find area of a circle whose circumference is 31.416 ft. OPERATION.–31.416 ft. ;- 3.1416 = 10 ft., diam. ; (10 ft.)? .7854 = 78.54 sq. ft., RULES. To find the area of a circle : area. 4. What is the area of a circular pond whose circumference is 200 chains ? Ans. 318.3 A. 5. The distance around a circular park is 14 miles. How many acres does it contain ? Ans. 114.59 A. 6. Find the area of the largest circle that can be drawn by using as a radius a string 20 in. long. 753. To find the diameter or circumference of a circle, when the area is given. 1. What is the diameter of a circle whose area is 1319.472 ? OPERATION.—1319.472-7854 = 1680 ; V1680 = 40.987 +, diameter. 2. What is the circumference of a circle whose area is 19.635? OPERATION—19.635 = 3.1416 = 6.25; V 6.25=2.5, radius ; 2.5 x 2 x 3.1416 = 15.708, circumference. Rule 1. Divide the area by .7854 and extract the square root of the quotient; the result is the diameter. 2. Divide the area by 3.1416 and extract the square root of the quotient; the result is the radius. The circumference is obtained by Art. (751, 1). Or, 3. Divide the area by .07958, and extract the square root of the quotient. 3. The area of a circular lot is 38.4846 square rods. What is its diameter ? Ans. 7 rods. 4. The area of a circle is 286.488 square feet. Required the diameter and the circumference. 754. To find the side of an inscribed square when the diameter of the circle is known. 1. What is the side of a square inscribed in a circle whose diameter is 6 rods? OPERATION.—62 = 2 = 18; V18 = 4.24 rd., side of 8quare. 2. The diameter of a circle is 200 feet. Find the side of the inscribed square. |