22 几何与三角

,第二级,第三级,第四级,第五级,New Words & Expressions:,angle 角 cube 立方体,arc 弧 curved line 曲线,major arc 优弧 cylinder 柱体,minor arc 劣弧 diameter 直径,architect 建筑师 dimention 维数，大小,breadth 宽度 endpoint 端点,chord 弦 equidistant 等距离的,circumference 周长 line segment 直线段,cone 圆锥 radius 半径,critical 临界的 pyramid 棱锥,2.2,几何与三角,Geometry and Trigonology,New Words & Expressions:,ray 射线 sphere 球，球面semicircle 半圆 surface 面，曲面solid 立体的，立体 thickness 厚度,Many leading institutions of higher learning have recognized that positive benefits can be gained by all who study this branch of mathematics.,2A Why study geometry?,许多居于领导地位的学术机构承认，所有学习这个数学分支的人都将得到确实的受益。,This is evident from the fact that they require study of geometry as a prerequisite to matriculation in those schools.,许多学校把几何的学习作为入学考试的先决条件，从这一点上可以证明。,Geometry is a science that deals with forms made by lines. A study of geometry is an essential part of the training of the successful engineer, scientist, architect, and draftsman.,几何是研究由线所组成的图形的科学。几何的学习是成功工程师，科学家，建筑师和制图员培训的重要部分。,The carpenter, machinist, stonecutter, artist, and designer all apply the facts of geometry in their trades. In this course the student will learn a great deal about geometric figures such as lines, angles, triangles, circles, and designs and patterns of many kinds.,木匠，机械师，采石者，艺术家和设计师在他们的职业中都应用几何的知识。在这门课程中，学生会学到大量几何图形，例如直线，角，三角形，圆以及许多种设计和模式。,One of the most important objectives derived from a study of geometry is making the student be more critical in his listening, reading and thinking. In studying geometry he is led away from the practice of blind acceptance of statements and ideas and is taught to think clearly and critically before forming conclusions.,学生通过几何的学习而达到的最主要目标是：在听，读，和思考时变得更加审慎。在学习几何的过程中，他们不再盲目地接受一些陈述和思想，而是在得出结论之前学会了清楚和审慎的思考。,1. Solids and planes. A solid is a three-dimensional figure. Common examples of solids are cube, sphere, cylinder, cone and pyramid.,2B Some geometrical terms,立体是一个三维图形，立体常见的例子是立方体，球体，柱体，圆锥和棱锥。,A cube has six faces which are smooth and flat. These faces are called plane surfaces or simply planes.,A plane surface has two dimensions, length and width. The surface of a blackboard or of a tabletop is an example of a plane surface.,立方体有6个面，都是光滑的和平的，这些面被称为平面曲面或者简称为平面。平面曲面是二维的，有长度和宽度，黑板和桌子上面的面都是平面曲面的例子。,2. Lines and line segments. We are all familiar with lines, but it is difficult to define,the term,. A line may be represented by the mark made by moving a pencil or pen across a piece of paper.,我们都熟悉直线，但是很难定义直线。可能可以用移动一支铅笔或者钢笔横穿一张纸留下的痕迹来描述直线。,A line may be considered as having only one dimension, length. Although when we draw a line we give it breadth and thickness, we think only of the length of the trace when considering the line.,直线可以被看做是一维的，即只有长度。尽管我们画一条直线的时候会赋予它宽度和厚度，但是当考虑直线的时候，只考虑痕迹的长度。,A point has no length, no width, and no thickness, but marks a position. We are familiar with such expressions as pencil point and needle point. We represent a point by a small dot and name it by a capital letter printed beside it, as “point A” in Fig. 2-2-1.,点没有长度，宽度和厚度，但是标记了一个位置。我们熟悉铅笔尖，针尖这样的表达。我们可以用一个小圆点来表示一个点，在它旁边用打印体大写字母来命名，如图2-2-1中的点A。,The line is named by labeling two points on it with capital letters or one small letter near it. The straight line extends infinitely far in two directions and has no ends. The part of the line between two points on the line is termed a line segment.,直线用大写字母标记它上面的两个点或者旁边的一个小写字母来命名。直线向两个方向无限延伸，没有终点。直线上两点间的部分被称为直线段。,A line segment is named the two end points. When no confusion result, the expression “line segment AB” is often replaced by segment AB or, simply, line AB.,直线段用两个端点命名。当不引起混淆的时候，“直线段AB” 通常被线段AB代替，或者简称直线AB。,There are three kinds of lines: the straight line, the broken line, and the curved line. A curved line or, simply, curve is a line no part of which is straight. A broken line is composed of joined, straight line segments, as ABCDE of Fig. 2-2-3.,有三种线：直线，折线和曲线。曲线是指其中没有任何部分是直的。折线是由连起来的直线段构成，如图2-2-3中的ABCDE.,3. Parts of a circle. A circle is a closed curve lying in one plane, all points of which are equidistant from a fixed point called the center. A line segment drawn from the center of the circle to a point on the circle is a radius of the circle.,平面上的闭曲线当其中每点到一个固定点的距离均相当时叫做圆。固定点称为圆心。连接圆心到圆周上点的直线段称为圆的半径。,经过圆心并且两个端点在圆周上的直线段被称为圆的直径。直径等于两个半径。连接圆周上两点的任意直线段被称为弦。,A diameter of a circle is a line segment through the center of the circle with endpoints on the circle. A diameter is equal to two radii. A chord is any line segment joining two points on the circle.,From the definition it should be apparent that a diameter is a chord. Any part of a circle is an arc, such that arc AE, which is denoted by .,从定义显然可见直径是弦。圆周的任何部分都是弧，例如,Points A and E divide the circle into minor arc AE and major arc ABE. A diameter divides a circle into two arcs termed semicircles, such as,and,.,The circumference is the length of a circle.,点A和E把圆周分为劣弧AE和优弧ABE。直径把圆周分为两个半圆，例如。圆周的长度即为周长。,One of the most important applications of trigonometry is the solution of triangles. Let us now take up the solutions of right angles. A triangle is composed of six parts, three sides and three angles. To solve a triangle is to find the parts not given.,2C Trigonometric function and solution of right triangles,三角学的重要应用之一是解三角形。现在我们接下来讲三角形的解。一个三角形由6个部分组成，三条边和三个角。解一个三角形就是要求出未知的部分。,A triangle may be solved if three parts (at least one of these is a side ) are given. A right triangle has one angle, the right angle, always given. Thus a right triangle can be solved when two sides, or one side and an acute angle, are given.,如果三角形的三个部分（其中至少有一个为边）为已知，则此三角形就可以解出。直角三角形的一只角，即直角，总是已知的。因此，如果它的两边，或一边和一锐角为已知，则此直角三角形可解。,作业：,P 31,2. 汉译英 (1), (3), (5)(8),3.英译汉,