MENSURATION 856. Mensuration is the process of finding the number of units in extension. Vertical. Horizontal. LINES. 857. A Straight Line is a line that does not change its direction. It is the shortest distance between two points. 858. A Curved Line changes its direction at every point. 859. Parallel Lines have the same direction; and being in the same plane and equally distant from each other, they can never meet. 860. A Horizontal Line is a line parallel either to the horizon or water level. 861. A Perpendicular Line is a straight line drawn to meet another straight line, so as to incline no more to the one side than to the other. A perpendicular to a horizontal line is called a vertical line. ANGLES. 862. An Angle is the difference in the direction of two lines proceeding from a common point, called the vertex. Angles are measured by degrees. (301.) 863. A Right Angle is an angle formed by two lines perpendicular to each other. 864. An Obtuse Angle is greater than a right angle. 865. An Acute Angle is less than a right angle. All angles except right angles are called oblique angles. PLANE FIGURES. 866. A Plane Figure is a portion of a plane surface bounded by straight or curved lines. 867. A Polygon is a plane figure bounded by straight lines. 868. The Perimeter of a polygon is the sum of its sides. 869. The Area of a plane figure is the surface included within the lines which bound it. (460.) A regular polygon has all its sides and all its angles equal. The altitude of a polygon is the perpendicular distance between its base and a side or angle opposite. A polygon of three sides is called a trigon, or triangle; of four sides, à tetragon, or quadrilateral; of five sides, a pentagon, etc. ооо Pentagon. Hexagon. Heptagon. Octagon. TRIANGLES. Nonagon. Decagon. 870. A Triangle is a plane figure bounded by three sides, and having three angles. 871. A Right-Angled Triangle is a triangle having one right angle. 872. The Hypothenuse of a rightangled triangle is the side opposite the right angle. Hypothenuse. Base. 90° Perpendicular. 873. The Base of a triangle, or of any plane figure, is the side on which it is supposed to stand. 874. The Perpendicular of a right-angled triangle is the side which forms a right angle with the base. 875. The Altitude of a triangle is a line drawn from the angle opposite perpendicular to the base. 1. The dotted lines in the following figures represent the altitude. 2. Triangles are named from the relation both of their sides and angles. 876. An Equilateral Triangle has its three sides equal. 877. An Isosceles Triangle has only two of its sides equal. 878. A Scalene Triangle has all of its sides unequal 879. An Equiangular Triangle has three equal angles (Fig. 1.) 880. An Acute-angled Triangle has three acute angles. (Fig. 2.) 881. An Obtuse-angled Triangle has one obtuse angle. (Fig. 3.) 882. The base and altitude of a triangle being given to find its area. 1. Find the area of a triangle whose base is 26 ft. and altitude 14.5 feet. OPERATION.-14.5 × 26÷2-1881 sq. ft. Or, 26 × feet, area. 14.5 =1881 square 2. What is the area of a triangle whose altitude is 10 yards and base 40 feet? RULE.-1. Divide the product of the base and altitude by 2. Or, 2. Multiply the base by one-half the altitude. Find the area of a triangle 3. Whose base is 12 ft. 6 in. and altitude 6 ft. 9 in. 4. Whose base is 25.01 chains and altitude 18.14 chains. 5. What is the cost of a triangular piece of land whose base is 15.48 ch. and altitude 9.67 ch., at $60 an acre? 6. At $.40 a square yard, find the cost of paving a triangular court, its base being 105 feet, and its altitude 21 yards? 7. Find the area of the gable end of a house that is 28 ft. wide, and the ridge of the roof 15 ft. higher than the foot of the rafters. 883. The area and one dimension being given to find the other dimension. 1. What is the base of a triangle whose area is 189 square feet and altitude 14 feet? OPERATION.-(189 sq. ft. x 2)+14=27 ft., base. 2. Find the altitude of a triangle whose area is 20 square feet and base 3 yards. RULE.-Double the area, then divide by the given dimension. Find the other dimension of the triangle 3. When the area is 65 sq. in. and the altitude 10 inches. 7. Paid $1050 for a piece of land in the form of a triangle, at the rate of $5 per square rod. If the base is 8 rd., what is its altitude? 884. The three sides of a triangle being given to find its area. 1. Find the area of a triangle whose sides are 30, 40, and 50 ft. OPERATION. 60-5010. (30+40+50)÷2 = 60; 60-30-30; 60-40 = 20; 60 × 30 × 20 × 10 = 600 ft., area. 2. What is the area of an isosceles triangle whose base is 20 ft., and each of its equal sides 15 feet? RULE. From half the sum of the three sides, subtract each side separately; multiply the half-sum and the three remainders together; the square root of the product is the area. 3. Find the area of a triangle whose sides are 25, 36, and 49 in. 4. How many acres in a field in the form of an equilateral tri angle whose sides each measure 70 rods ? 5. The roof of a house 30 ft. wide has the rafters on one side 20 ft. long, and on the other 18 ft. long. How many square feet of boards will be required to board up both gable ends? 885. The following principles relating to right-angled triangles have been established by Geometry: PRINCIPLES.-1. The square of the hypothenuse of a right-angled triangle is equal to the sum of the squares of the other two sides. 2. The square of the base, or of the perpendicular, of a right-angled triangle is equal to the square of the hypothenuse diminished by the square of the other side. 886. To find the hypothenuse. 1. The base of a right-angled triangle is 12, and the perpendicular 16. What is the length of the hypothenuse? OPERATION.—122 +162 = 400 (PRIN. 1). 400= 20, hypothenuse. 2. The foot of a ladder is 15 feet from the base of a building, and the top reaches a window 36 feet above the base. What is the length of the ladder ? RULE.—Extract the square root of the sum of the squares of the base and the perpendicular; the result is the hypothenuse. 3. If the gable end of a house 40 ft. wide is 16 ft. high, what is the length of the rafters? 4. A park 25 chains long and 23 chains wide has a walk running through it from opposite corners in a straight line. What is the length of the walk? What is the 5. A room is 20 ft. long, 16 ft. wide, and 12 ft. high distance from one of the lower corners to the opposite upper corner? 887. To find the base or perpendicular. 1. The hypothenuse of a right-angled triangle is 35 feet, and the perpendicular 28 feet. Find the base. OPERATION.—35a — 28a = 441 (PRIN. 2). √44121 ft., base. |
