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2. Draw a figure to illustrate part of the infinite pattern that can be derived from a given equilateral triangle ABC by applying all the vectors con- sisting of an integral multiple of AB plus an integral multiple of AC.

2. Draw a figure to illustrate part of the infinite pattern that can be derived from a given equilateral triangle ABC by applying all the vectors con- sisting of an integral multiple of AB plus an integral multiple of AC.

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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Answer just #2 please proof
9:45 1 LTE A aproged.pt positions of the circle meet at B and C, these are two vertices of the desired rectangle ABCD, whose sides AB and DC are equal and parallel to a. 82 TRANSFORMATIONS Figure 4.1D EXERCISES 1. In AABC (Figure 4.1E) “inscribe" a line segment equal and parallel to the given segment a. 2. Draw a figure to illustrate part of the infinite pattern that can be derived from a given equilateral triangle ABC by applying all the vectors con- sisting of an integral multiple of AB plus an integral multiple of AC. Figure 4.1E 4.2 Rotation Another kind of transformation that preserves distance is rotation. Here the entire plane is turned about some point through a given angle. Thus the size and shape of any figure are kept invariant, but its points all move along arcs of concentric circles. The center (which may or may not “belong" to the figure being rotated) is the only point that remains fixed. As an example of the use of a rotation, let us consider AABC (Figure 4.2A) with equilateral triangles BPC, CQA, ARB erected (externally) on the three sides. After drawing the lines BQ and Cr, which meet at F, we observe that a rotation through 60° about A takes AARC into AA BQ. Hence ZRFB = 60° and RC = BQ. Similar reasoning shows that PA = CR. Thus AP = BQ = CR. ROTATION 83 Moreover, since ZRFB = 60° = ZRAB and 2CFQ = 60° = 2CAQ, the quadrangles A RBF and CQAF are cyclic; and since ZBFC = 120° while 2CPB = 60 PEPi drangle. Therefore the
Transcribed Image Text:9:45 1 LTE A aproged.pt positions of the circle meet at B and C, these are two vertices of the desired rectangle ABCD, whose sides AB and DC are equal and parallel to a. 82 TRANSFORMATIONS Figure 4.1D EXERCISES 1. In AABC (Figure 4.1E) “inscribe" a line segment equal and parallel to the given segment a. 2. Draw a figure to illustrate part of the infinite pattern that can be derived from a given equilateral triangle ABC by applying all the vectors con- sisting of an integral multiple of AB plus an integral multiple of AC. Figure 4.1E 4.2 Rotation Another kind of transformation that preserves distance is rotation. Here the entire plane is turned about some point through a given angle. Thus the size and shape of any figure are kept invariant, but its points all move along arcs of concentric circles. The center (which may or may not “belong" to the figure being rotated) is the only point that remains fixed. As an example of the use of a rotation, let us consider AABC (Figure 4.2A) with equilateral triangles BPC, CQA, ARB erected (externally) on the three sides. After drawing the lines BQ and Cr, which meet at F, we observe that a rotation through 60° about A takes AARC into AA BQ. Hence ZRFB = 60° and RC = BQ. Similar reasoning shows that PA = CR. Thus AP = BQ = CR. ROTATION 83 Moreover, since ZRFB = 60° = ZRAB and 2CFQ = 60° = 2CAQ, the quadrangles A RBF and CQAF are cyclic; and since ZBFC = 120° while 2CPB = 60 PEPi drangle. Therefore the
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