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Angles and triangles
Published in John Bird, Bird's Basic Engineering Mathematics, 2021
In triangle DEF,∠E=180∘−50∘−60∘=70∘ Hence, triangles ABC and DEF are similar, since their angles are the same. Since corresponding sides are in proportion to each other, ad=cfi.e.a4.42=12.05.0
Scale Modeling
Published in Diane P. Michelfelder, Neelke Doorn, The Routledge Handbook of the Philosophy of Engineering, 2020
Note that the criterion of similarity in use here is objective. In spite of the fact that the situations compared have aesthetic aspects and that human cognition is involved in apprehending the two triangular figures associated with the two physical situations, the criterion of geometrical similarity between the two triangular figures indicated in Figure 29.4 is completely objective. The question of whether two plane triangles are geometrically similar is settled here by the fact that the two triangles are right triangles and the angle at the top of the tree and the angle at the top of the student’s head are formed by rays of the sun in the sky hitting them at the same angle. That angle need not even be known in order to conclude that the triangles indicated in Figure 29.4 are similar triangles. The reasoning from geometric similarity is objective, too, i.e., the consequence of the fact that these two triangles have the same shape, i.e., are geometrically similar, is that ratios between corresponding sides are the same. The reasoning from geometric similarity is straightforward reasoning according to the methods of Euclidean geometry. In Euclidean geometry, what’s similar are two dimensional closed curves (figures), or, if three dimensional, solid figures.
Angles and triangles
Published in John Bird, Basic Engineering Mathematics, 2017
In triangle DEF,∠E=180∘-50∘-60∘=70∘ $ DEF , \angle E = 180^{\circ } - 50^{\circ } - 60^{\circ } = 70^{\circ } $ Hence, triangles ABC and DEF are similar, since their angles are the same. Since corresponding sides are in proportion to each other, ad=cfi.e.a4.42=12.05.0 $$ \begin{aligned} \frac{a}{d} = \frac{c}{f}\quad \text{ i.e.}\quad \frac{a}{4.42} = \frac{12.0}{5.0} \end{aligned} $$
Angle-side properties of polygons inscribable in an ellipse
Published in International Journal of Mathematical Education in Science and Technology, 2022
Jay Jahangiri, Ruti Segal, Moshe Stupel
For the integers , consider the ellipse with centre C so that its n diameters intersect the ellipse at the vertices (Figure 1). Then for and we have the following two identities .Since the two triangles and are congruent, their corresponding sides are also congruent. So Therefore For the second part, notice that each diagonal divides the corresponding interior angles and of the 2n-gon into two smaller angles so that and
Pre-service mathematics teachers’ semiotic transformation of similar triangles: Euclidean geometry
Published in International Journal of Mathematical Education in Science and Technology, 2022
The above dialogue revealed that the participant was unable to visualize the conversion from one semiotic representation to another in a different register. We observe the participant not making sense of the geometric information given about the two triangles; one being an enlargement of the other while the other is a reduction of the other. However, the progress of the interview continued as follows: A: Let us observe the angles of the two triangles to verify if they are equal. Then by illustration through observation, based on geometric rules, P= S, Q = T and R = V. Do you agree that the two triangles are equal?T16: Yes.A: Now what does it mean when the corresponding angles of a triangle are equal and proportional?T16: SilentA: Alright, let us use an example; if the sides of the first triangle are 2, 3, and 4 then if you multiply by a factor of 2, all the three sides will be multiplied by a factor of 2 to get an enlarged triangle, 4, 6 and 8. If you multiply by a factor of 3, then the new triangle could be 6, 9, and 12, which means that if you divide the first triangle by an enlarged triangle you will get the same answer, for instance = ; = and = .A: So, when two triangles are similar, we know that the corresponding sides are proportional.T16: Ok ma.A: Even if we do not know the sizes of the triangles, we can divide we get the same answer as and we get the same answer as.T16: Ok.A: Proportionality means that the ratio of the sides is the same.