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Introduction to Linear Algebra
Published in Timothy Bower, ®, 2023
As illustrated in figure 5.5, a rhombus is any parallelogram whose sides are the same length. Let us use the properties of dot products to show that the diagonals of a rhombus are perpendicular.
Sketcher Workbench
Published in Kaushik Kumar, Chikesh Ranjan, J. Paulo Davim, Understanding CATIA, 2021
Kaushik Kumar, Chikesh Ranjan, J. Paulo Davim
The parallelogram is defined by three corner points and opposite sides parallel to each other. For drawing a parallelogram, select Parallelogram tool from Predefine toolbar as shown in Figure 2.10. Click the geometry area to indicate the first corner of the parallelogram followed by the second click to complete the first side. Finally, click a location to indicate the diagonally-opposite corner as shown in Figure 2.13.
Triangles and quadrilaterals
Published in James Kidd, Ian Bell, Maths for the Building Trades, 2014
In a parallelogram the sides that are opposite each other are of equal length and are parallel to each other (Figure 10.23). The angles that are opposite to each other are also equal. Providing one of the angles is known, the other three can be found. The diagonals AD and BC bisect each other and the parallelogram to form two congruent triangles. The area of a parallelogram can be found by multiplying the base by the perpendicular height (this means at right angles to the base).
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
None of the definitions given above include superfluous information. For instance, in TR-4 (Re), it is sufficient to state ‘a parallelogram with one right angle’ rather than ‘a parallelogram with four right angles’ to define a rectangle because, in Euclidean geometry, it is possible to prove that in a parallelogram if there is at least one right angle, then there are four right angles. Exemplary non-minimal definitions are presented as follows: US-5 (Re): A parallelogram with four right anglesTR-2 (Re): A parallelogram in which all angles are right anglesUS-2 (S): A parallelogram with four congruent sides and four right anglesTR-4 (S): A rectangle with all sides congruentTR-7 (Rh): A parallelogram whose sides are equal in lengthUS-1 (Rh): A parallelogram in which diagonals are perpendicular and bisect each pair of opposite angles
Toward a unified account of definitions in mathematics education research: a systematic literature review
Published in International Journal of Mathematical Education in Science and Technology, 2023
Hermund André Torkildsen, Tore Alexander Forbregd, Eivind Kaspersen, Trygve Solstad
To mitigate the obstruction of learning due to minimal definitions, Zazkis and Leikin (2008) introduced the term barely-not-minimal, as definitions that use appropriate terminology and contain necessary and sufficient conditions without being minimal (Johnson et al., 2014, p. 288) (c.f. essentiality). For example, a rectangle can be defined as a parallelogram with four right angles. This definition is not minimal since it contains a theorem. It is sufficient to require a parallelogram to have a single right angle, from which it follows that all the angles are right angles. However, in certain circumstances, it is minimal enough.
Focusing attention on auxiliary lines when introduced into geometric problems
Published in International Journal of Mathematical Education in Science and Technology, 2019
In all three approaches, the presence of midpoints affords concretization of the definition of a triangle midsegment. The diagonal of the given quadrilateral is introduced first and divides that quadrilateral into two triangles, thus allowing the use of the property of a triangle midsegment (equals half of the side it is parallel to AC||EF||HG and ½ AC = EF = HG). Note that in case 13a, there are two such partitions: AC and BD. The quadrilateral EFGH is proved to be a parallelogram either due to the property of opposite sides or by the definition of a parallelogram.