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Basic Concepts of Descriptive Geometry
Published in Ken Morling, Stéphane Danjou, Geometric and Engineering Drawing, 2022
The theorem of Pythagoras says that ‘in a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides’. When this theorem is shown pictorially, it is usually illustrated by a triangle with squares drawn on the sides. This tends to be a little misleading since the theorem is valid for any similar plane figures (Figure 5.31).
Marking out and measuring
Published in Andrew Livesey, Alan Robinson, The Repair of Vehicle Bodies, 2018
This is the most common tool for testing squareness and is used for internal and external testing to check whether work is square (Figure 6.5). It can also be used for setting out lines at right angles to an edge or surface. A try square consists of a stock and a blade, and these may be made separately and joined together, or the whole square may be formed from a single piece of metal. Squares are of cast steel, hardened and tempered and ground to great accuracy. The size of the square is the inside length of the blade. Engineers’ squares are made with a groove cut in the stock where the blade enters the stock; this allows for any burr on the edge of the metal being tested.
Inclined Cuboids
Published in Craig Attebery, The Complete Guide To Perspective Drawing, 2018
A third dimension is extruded from this inclined plane. All the corners are right angles. The thickness projects 90° from the inclined plane. To draw the thickness, another auxiliary vanishing point is needed, a point 90° from the original auxiliary vanishing point. Thus, the first step is to locate the point of true angles—the measuring point. At the measuring point, draw a 90° angle from the original auxiliary vanishing point. There are now two auxiliary vanishing points that are 90° apart (Figure 10.2).
Parallel curves
Published in International Journal of Mathematical Education in Science and Technology, 2022
Richard Dexter Sauerheber, Tony Stewart
Figure 1 shows the relationship between parallel lines, where vectors are drawn from one line to corresponding points on the parallel line where the connecting vector is perpendicular to both lines at the same time. Each connecting vector is the same length because the lines are themselves parallel and cannot intersect. Parallel lines must share the same derivative while being shifted in the Euclidean plane. Geometry courses routinely and correctly teach proofs indicating that parallel lines have linear transversals that must form congruent alternate interior angles, congruent alternate exterior angles, and pairs of interior angles and exterior angles on the same side of the transversal that must be supplementary ( = 180°) (Ratti and McWaters, 2010). This also means that any segment or vector Eextending from one line to the other that is perpendicular to one line must also be perpendicular to the other parallel line. All perpendicular segments intersecting any two parallel lines must always form only right angles and must be of equal length. If not, then the lines are not parallel.
Recognizing the Correlation of Architectural Drawing Methods between Ancient Mathematical Books and Octagonal Timber-framed Monuments in East Asia
Published in International Journal of Architectural Heritage, 2023
The Pythagorean theorem states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the two other sides. Accordingly, the Song Yingzao Fashi’s Building Standards adopt the method of finding the hypotenuse from a right-angled triangle. Thus, the base length of the deleted triangle for drawing an octagon from a square is slightly shorter than the side length of the octagon (Liu 2014, 66). This discrepancy indicates that the theorem does not cover the equilateral octagons defined by modern mathematics [Figure 4].
Middle school students’ reasoning with regards to parallelism and perpendicularity of line segments
Published in International Journal of Mathematical Education in Science and Technology, 2023
Examples have both critical and non-critical attributes (Hershkowitz, 1989). A concept definition is linked to critical attributes of a mathematical concept (Ulusoy, 2019). Non-critical attributes, on the other hand, can be acquired by reasoning about a concept's formal definition, which contains just necessary and sufficient defining criteria (Tsamir et al., 2008). Non-critical attributes are only found in prototypical examples of the concept (Hershkowitz, 1989; Ulusoy, 2021). For instance, perpendicular lines are two lines that intersect at a right angle. This concept includes a number of critical attributes, some of them are explicit and others are implicit. (Gal & Vinner, 1997) mention these critical attributes as follows: ‘(1) two lines (sometimes two segments or one line and one segment), (2) the lines intersect, (3) a right angle is formed at the point of intersection, (4) there are three more angles at the point of intersection, (5) these three other angles are also right angles, and (6) a right angle is a 90° angle’ (p.282). Therefore, whereas all examples of a concept contain the entire set of critical attributes, only some prototypical examples of the concept contain non-critical attributes of the concept (e.g. Ulusoy, 2019). The prototypes are usually the subset of examples that had the longest list of attributes all the critical attributes of the concept and those specific (noncritical) attributes that had strong visual characteristics’ (Hershkowitz, 1990, p. 82). For example, two parallel line segments are mostly represented in vertical or horizontal orientations in lessons or textbooks. However, when two parallel line segments are represented in an inclined orientation, this example shows parallel line segments in a non-prototypical orientation. In mathematics education, students are expected to use only critical attributes when generating and identifying examples of geometric concepts because it is necessary to recognize critical attributes to provide the correct identification of a figure (van Hiele & van Hiele, 1958). However, according to the findings of relevant studies, students do not have the expected degree of knowledge of geometric ideas (Burger & Shaughnessy, 1986; Gutiérrez & Jaime, 1999; Hannibal, 1999) due to the dominancy of prototypical concept images in students’ minds. As a result, when students encounter prototypical examples of a geometric concept, they make their judgments based on their poorest concept images developed on visual characteristics of prototypes (e.g. Hershkowitz, 1989, 1990; Fujita, 2012). Thus, they can identify some examples of the concept as non-examples or they can identify some non-examples as examples of the concept.