Geometry for Elementary School/Exercise 1
You have gone through the entire section about the basic concepts of geometry. It is about time you test yourself. Note that this exercise covers only knowledge questions and no sums are required. If you need a detailed solution, please see the answers.
Section A: Points, lines and angles
Part one: True or False
- Points are infinitely small.
- Lines have a fixed length.
Part two: Multiple choice questions
- Which of the following facts about line segments is correct?
- Any two lines segments must intersect.
- Line segments are lines which go on forever in both directions
- Line segments have a finite length because they have midpoints
- When two line segments intersect, the point where they cross is called the point of intersection
Section B: Plane and solid figures
We've went over this, right? But it is very important to go over while doing Exercise I. Plane figures, such as a circle, are flat, like a piece of paper. Solids are shapes you can actually touch. You also need to think about the texture. Texture is how an object feels. Here is an example, I have foam in my hand. It feels squishy and foamy.
Section C: Symmetry, transformation and coordinates
Symmetry (from the Greek: "συμμετρεῖν" = to measure together), generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection. The second meaning is a precise and well-defined concept of balance or "patterned self-similarity" that can be demonstrated or proved according to the rules of a formal system: by geometry, through physics or otherwise.
Although the meanings are distinguishable in some contexts, both meanings of "symmetry" are related and discussed in parallel.
The "precise" notions of symmetry have various measures and operational definitions. For example, symmetry may be observed:
with respect to the passage of time; as a spatial relationship; through geometric transformations such as scaling, reflection, and rotation; through other kinds of functional transformations; and as an aspect of abstract objects, theoretic models, language, music and even knowledge itself.
This article describes these notions of symmetry from four perspectives. The first is that of symmetry in geometry, which is the most familiar type of symmetry for many people. The second perspective is the more general meaning of symmetry in mathematics as a whole. The third perspective describes symmetry as it relates to science and technology. In this context, symmetries underlie some of the most profound results found in modern physics, including aspects of space and time. Finally, a fourth perspective discusses symmetry in the humanities, covering its rich and varied use in history, architecture, art, and religion.
The opposite of symmetry is asymmetry.
In mathematics, a transformation could be any function mapping a set X on to another set or on to itself. However, often the set X has some additional algebraic or geometric structure and the term "transformation" refers to a function from X to itself that preserves this structure.
Examples include linear transformations and affine transformations, rotations, reflections and translations. These can be carried out in Euclidean space, particularly in dimensions 2 and 3. They are also operations that can be performed using linear algebra, and described explicitly using matrices.
In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element. The order of the coordinates is significant and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in 'the x-coordinate'. In elementary mathematics the coordinates are taken to be real numbers, but in more advanced applications coordinates can be taken to be complex numbers or elements of a more abstract system such as a commutative ring. The use of a coordinate system allows problems in geometry to be translated into problems about numbers and vice versa; this is the basis of analytic geometry.
An example in everyday use is the system of assigning longitude and latitude to geographical locations. In physics, a coordinate system used to describe points in space is called a frame of reference.
Section D: Integrated and miscellaneous questions
Integration may refer to:
In sociology and economy:
Social integration Racial integration, refers to social and cultural behavior; in a legal sense, see desegregation Economic integration Educational integration of students with disabilities Regional integration American Studies Integration, the study of pillars kind of routine Horizontal integration and vertical integration, in microeconomics and strategic management, refer to a style of ownership and control Integration clause, in a contract, a term used to declare the contract the final and complete understanding of the parties Integrated production
Integration, in mathematics, a fundamental concept of calculus—the operation of calculating the area between the curve of a function and the x-axis Indefinite integration, in calculus, the process of calculating antiderivatives—the opposites of derivatives (a.k.a. "antidifferentiation") In numerical analysis and in signal processing, a time series can be "integrated" numerically by various step-by-step means, including autoregressive integrated moving average and the Runge-Kutta methods.
In Electronics Engineering:
Integrated circuit, an electronic circuit whose components are manufactured in one flat piece of semiconductor material Systems integration, the engineering practices and procedures for assembling large and complicated systems from less-complicated units, especially subsystems In electronic signal processing, radars, sonars, and astronomy, integration is the process of taking multiple copies of a weak signal and adding them together to form detectable signals. Special vacuum tubes called "integration tubes" were devised and used in radar systems.
Pre-integration complex or retroviral integration, in biology In animation, to combine 2D, 3D graphics, motion capture, or live-action footage together