An isosceles triangle, Fig. 50, is one, two of whose sides are equal. Fig. 49. Equilateral 4 A Fig. 50. Isosceles Fig. 51. Scalene Triangle The base of a triangle is the lowest side; it is the side upon which the triangle is supposed to stand. Any side may, however, be taken as the base. In an isosceles triangle, the side which is not one of the equal sides is usually considered as the base. The altitude of a triangle is the perpendicular drawn from the vertex to the base. Quadrilaterals. A quadrilateral is a polygon bounded by four straight lines, as Fig. 52. The diagonal of a quadrilateral is a straight line joining two opposite vertices. Trapezium. A trapezium is a quadrilateral, no two of whose sides are parallel. Trapezoid. A trapezoid is a quadrilateral having two sides Fig. 52. Quadrilateral Fig. 53. Trapezoid Fig. 54. Parallelogram parallel, Fig. 53. The parallel sides are called the bases and the perpendicular distance between the bases is called the altitude. Parallelogram. A parallelogram is a quadrilateral whose opposite sides are parallel, Fig. 54. There are four kinds of parallelograms: rectangle, square, rhombus, and rhomboid. The rectangle, Fig. 55, is a parallelogram whose angles are right angles. The square, Fig. 56, is a parallelogram all of whose sides are equal and whose angles are right angles. The rhombus, Fig. 57, is a parallelogram whose sides are equal but whose angles are not right angles. Fig. 55. Rectangle Fig. 56. Square Fig. 57. Rhombus The rhomboid is a parallelogram whose adjacent sides are unequal, and whose angles are not right angles. CIRCLES 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, Fig. 58. A diameter of a circle is a straight line drawn through the center, terminating at both ends in the circumference, Fig. 59. A radius of a circle is a straight line joining the center with the circumference. All radii of the same circle are equal and their length is always one-half that of the diameter. An arc An arc is any part of the circumference of a circle. equal to one-half the circumference is called a semi-circumference, and an arc equal to one-quarter of the circumference is called a quadrant, Fig. 60. A quadrant may mean the arc or angle. A chord, Fig. 61, is a straight line which joins the extremities of an arc but does not pass through the center of the circle. A secant is a straight line which intersects the circumference in two points, Fig. 61. A segment of a circle, Fig. 62, is the area included between an arc and a chord. A sector is the area included between an arc and two radii drawn to the extremities of the arc, Fig. 62. A tangent is a straight line which touches the circumference at only one point, called the point of tangency or contact, Fig. 59. Concentric circles are circles having the same center, Fig. 63. An inscribed angle is an angle whose vertex lies in the circumference and whose sides are chords. It is measured by one-half the intercepted arc, Fig. 64. A central angle is an angle whose vertex is at the center of the circle and whose sides are radii, Fig. 65. An inscribed polygon is one whose vertices lie in the circumference and whose sides are chords, Fig. 66. MEASUREMENT OF ANGLES To measure an angle, take any convenient radius and describe an arc with the center at the vertex of the angle. The portion of the arc included between the sides of the angle is the measure of the angle. If the arc has a constant radius, the greater the divergence of the sides, the longer will be the arc. If there are several arcs drawn with the same center, the intercepted arcs will have different lengths but they will all be the same fraction of the entire circumference. 4 5 105 90 75 120 135 45 150 30 15 165 In order that the size of an angle or arc may be stated without saying that it is a certain fraction of a circumference, the circumference is divided into 360 equal parts called degrees, Fig. 67. Thus, it may be said that a certain angle contains 45 degrees, i.e., it is of a circumference. In order to obtain accurate measurements each degree is divided into 60 equal parts called minutes and each minute into 60 equal parts called seconds. 180 Fig. 67. Angular Measurement SOLIDS A solid has three dimensions-length, breadth, and thickness. The most common forms of solids are polyhedrons, cylinders, cones, and spheres. POLYHEDRONS A polyhedron is a solid bounded by planes. The bounding planes are called faces and their intersections are called edges. The intersections of the edges are called vertices. A polyhedron having four faces is called a tetrahedron; one having six faces, a hexahedron; one having eight faces, and octahedron, Fig. 68; one having twelve faces, a dodecahedron, etc. Prisms. A prism is a polyhedron having two opposite faces, called bases, which are equal and parallel, and other faces, called Fig. 68. Octahedron Fig. 69. Prism Fig. 70. Right Prism lateral faces, which are parallelograms, Fig. 69. The altitude of a prism is the perpendicular distance between the bases. The area of the lateral faces is called the lateral area. Prisms are called triangular, rectangular, hexagonal, etc., according to the shape of the bases. Further classifications are as follows: Fig. 71. Parallelopiped Fig. 72. Rectangular Paral- Fig. 73. Truncated A right prism is one whose lateral faces are perpendicular to the bases, Fig. 70. A regular prism is a right prism having regular polygons for bases. Parallelopiped. A parallelopiped is a prism whose bases are parallelograms, Fig. 71. If all the edges are perpendicular to the bases, it is called a right parallelopiped. A rectangular parallelopiped is a right parallelopiped whose bases and lateral faces are rectangles, Fig. 72. A cube is a rectangular parallelopiped all of whose faces are squares. A truncated prism is the portion of a prism included between the base and a plane not parallel to the base, Fig. 73. Pyramids. A pyramid is a polyhedron whose base is a polygon and whose lateral faces are triangles having a common vertex called the vertex of the pyramid. Fig. 74. Pyramid Fig. 75. Regular Pyramid Fig. 76. Frustum of The altitude of the pyramid is the perpendicular distance from the vertex to the base. Pyramids are named according to the kind of polygon forming the base, viz, triangular, quadriangular, Fig. 74. pentagonal, Fig. 75, hexagonal. |