MENSURATION. 702. Mensuration is the process of finding the number of units in extension. LINES. 703. A Straight Line is a line that does not change its direction. It is the shortest distance between two points. 704. A Curved Line changes its direc tion at every point. 705. Parallel Lines have the same direc: tion; and being in the same plane and equally distant from each other, they can never meet. 706. A Horizontal Line is a line parallel to the horizon or water level. 707. 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. 708. Oblique Lines approach each other, and will meet if sufficiently extended. Vertical. Horizontal. V ANGLES. 709. An Angle is the opening between two lines that meet each other in a common point, called the vertex, Angles are measured by degrees (301). 710. A Right Angle is an angle formed by two lines perpendicular to each other. 711. An Obtuse Angle is greater than 9 right angle. 712. An Acute Angle is less than a right angle. All angles except right angles are called oblique angles. 713. A Plane Figure is a portion of a plane surface bounded by straight or curved lines. 714. A Polygon is a plane figure bounded by straight lines. 715. The Perimeter of a polygon is the sum of its sides. 716. The Area of a plane figure is the surface included within the lines which bound it (282). A regular polygon has all its sides and all its angles equal. A polygon of three sides is called a trigon, or triangle ; of four sides, a tetragon, or quadrilateral ; of five sides, a pentagon, etc. Perpendicular. Hypothenuse. 90° 717. A Triangle is a plane figure bounded by three sides, and having three angles. 718. A Right-Angled Triangle is a triangle having one right angle. 719. The Hypothenuse of a rightangled triangle is the side opposite the right angle. Base. 720. The Base of a triangle, or of any plane figure, is the side on which it may be supposed to stand. 721. The Perpendicular of a right-angled triangle is the side which forms a right angle with the base. 722. The Altitude of a triangle is a line drawn perpendicular to the base from the angle opposite. 1. The dotted vertical lines in the figures represent the altitude. 2. Triangles are named from the relation both of their sides and angles. 723. An Equilateral Triangle has its three 724. An Isosceles Triangle has only two ox 725. A Scalene Triangle has all of its sides ut. 726. An Equiangular Triangle has three equal angles. (Fig. 1.) 727. An Acute-angled Triangle has three acute angles. (Fig. 2.) 728. An Obtuse-angled Triangle has one obtuse angle. (Fig. 3.) PROBLEMS. 729. 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 ft. 14.5 OPERATION.–14.5 x 26+2=1884 sq. ft. Or, 26 x -=1884 sq. ft., area. 2 2. What is the area of a triangle whose altitude is 10 yd. and base 40 ft. ? Ans, 600 sq. ft. 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. A. 42s.f. 4. Whose base is 25.01 ch. and altitude 18.14 ch. 22, 6:11, 5. What will be 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, what is the cost of paving a triangular court, its base being 105 ft. and altitude 21 yards ? Ans. $147. 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. 30. The area and one dimension being given to find je other dimension. 1. What is the base of a triangle whose area is 189 sq. ft. and altitude 14 ft. ? OPERATION.-189 sq. ft. x 2+14 = 27 ft., base. 2. Find the altitude of a triangle whose area is 207 sq. ft. and base 3 yards. Ans. 41 ft. RULE. Double the area, then divide by the given dimension. 7. Paid $1050 for a piece of land in the form of a triangle, at the rate of $51 per square rod. If the base is 8 rd., what is its altitude ? Ans. 50 rods. 731. 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.–(30+40+50)-2 = 60; 60—30=30; 60-40=20; 60-50 - 10. V60 x 30 x 20 x 10 = 600 ft., area. 2. What is the area of an isosceles triangle whose base is 20 ft., each of its equal sides 15 ft. ? Ans. 111.85 sq. ft. 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. How many acres in a field in the form of an equilateral triangle whose sides measure 70 rods? Ans. 13 A. 41.76 P. 4. 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 ? 732. The following principles relating to right-angled triangles have been established by Geometry: 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. 733. 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+16° = 400 (Prin. 1). V400 = 20, hypothenuse. 2. The foot of a ladder is 15 ft. from the base of a building, and the top reaches a window 36 ft. above the base. What is the length of the ladder ? Ans. 39 ft. RULE. Extract the square root of the sum of the squares of the base and the perpendicular; and the result is the hypothenuse. 3. If the gable end of a house 40 feet wide is 16 ft. high, what is the length of the rafters ? 4. A park 25 ch. long and 23 ch. wide has a walk running through it from opposite corners in a straight line. What is the length of the walk? Ans. 33.97 ch. t. 5. A room is 20 ft. long, 16 ft. wide, and 12 ft high. What is the distance from one of the lower corners to the opposite upper corner ? Ans. 28 ft. 3.36 in. 734. To find the base or perpendicular. 1. The hypothenuse of a right-angled triangle is 35 ft., and the perpendicular 28 ft. What is the base ? OPERATION. —-352 — 289 = 441 (Prin. 2). V 441 = 21, base. |