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Published in Fadi Al-Turjman, Cognitive Sensors and IoT, 2017
For a square of side L, the diagonals intersect at the center of the square, and are perpendicular bisectors of each other. Using Pythagoras’s theorem [24], we know that the length of the diagonal which forms the hypotenuse of the isosceles triangle formed by the two sides of the square, can be found as (L2+L2)=2∗L. Thus the center of a square is equidistant from each of its vertices and has a length L/2, as shown in Figure 3.3(a).
Triangles and quadrilaterals
Published in James Kidd, Ian Bell, Maths for the Building Trades, 2014
A square (Figure 10.26) is a rectangle with all its sides equal in length. The two diagonals AD and CB are equal in length and bisect each other. The diagonals intersect at 90° and bisect the square to form congruent triangles. The area of a square = L2, where L is the length of one side of the square.
Cognitive Development
Published in Krystina Castella, Designing for Kids, 2018
Deductive reasoning is when you look at an end result and then figure out – or deduce – how it came about. Deductive reasoning is the process of reasoning from the general to the specific. For example, since all squares are rectangles, and all rectangles have four sides, all squares have four sides.46
A comparison of mathematical features of Turkish and American textbook definitions regarding special quadrilaterals
Published in International Journal of Mathematical Education in Science and Technology, 2019
Mathematically correct structural definitions include properties (conditions) that are both necessary and sufficient [18,21,67]. Thus, such definitions describe the components of a special quadrilateral appropriately. On the other hand, mathematically incorrect structural definitions include either necessary but not sufficient conditions (insufficient conditions), sufficient but not necessary conditions (unnecessary conditions), or neither necessary nor sufficient conditions. Consider, for example, the following structural definitions: 1) a parallelogram is a quadrilateral with both pairs of opposite sides parallel, 2) a kite is a quadrilateral with perpendicular diagonals, 3) a rhombus is a quadrilateral with congruent sides and angles, and 4) a square is a geometric solid with congruent sides and four right angles. Definition 1 provides necessary and sufficient conditions for a parallelogram and thus, it is a mathematically correct definition. In Definition 2, the ‘perpendicular diagonals’ property is not sufficient to define a kite because it allows for the construction of a non-special quadrilateral whose diagonals are perpendicular. Thus, it can be considered as an incorrect definition. In Definition 3, the ‘congruent angles’ condition is not necessary, because it does not apply to prototypical rhombi. Thus, it is an incorrect definition. In Definition 4, being a ‘geometric solid’ includes internal inconsistency in that quadrilaterals are two-dimensional objects whereas geometric solids are three-dimensional. Thus, this definition includes neither necessary nor sufficient conditions for a square and thus, it is an incorrect definition.
Reconsidering a proportional system of timber-frame structures through ancient mathematics books: a case study on the Muryangsujŏn Hall at Pusŏksa Buddhist Monastery
Published in Journal of Asian Architecture and Building Engineering, 2019
Furthermore, Liu Hui states that the Yellow Emperor transforms and extends the cosmic principle of the trigrams tremendously to solve practical problems, such as divination, regulation of the calendar, and harmonization of the musical scale. The prefatory chapter “Kanxiang” of the Yingzao Fashi, as has been noted earlier, mentions various phrases that relate to the length of figures and buildings. The parts of rectangles are as follows: “If the lateral side of a square is one hundred in length, its diagonal line across the square is one hundred and forty-one in length (方一百其斜一百四十有一).” In this passage, the square root of 2 (√2), or the length of a diagonal line across a square, is notable. The simplest mathematical form of √2 is an equilateral triangle. Assuming that the equilateral triangle’s base and height measure 1 at right angles, the length of the hypotenuse of the triangle is defined as the square root of 2 or 1.414 when expressed as a numerical value in modern mathematics. A square results from combining two equilateral triangles. The method prescribed in the Yingzao Fashi to determine the length of figures is based on the selection of the Pythagorean Theorem.9Liu Hui describes the ratio of diagonal length in the octagonal geometrical construction method given in the Jiuzhang Suanshu, and argues that the error of the square root of 2 (√2) is approximately 1% (Liu 2014). The formative ideas of the ancients, in consort with the composition principle of circles and squares, which are embedded in the Zhoubi Suanjing and the Jiuzhang Suanshu, permeate the contents of the Yingzao Fashi.