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Given an acute-angled triangle ABC, construct a square with one side lying on BC while the other two vertices lie on CA and AB, respectively.

Given an acute-angled triangle ABC, construct a square with one side lying on BC while the other two vertices lie on CA and AB, respectively.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.2: Graphs Of Equations
Problem 22E
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9:46 1 LTE A aproged.pt OA' = - kOA OB' – kOB, and DILATATION 95 as in Figure 4.7B. Remembering that parallel lines cut transversals into proportional segments, we easily deduce that C' lies on OC; in fact, Varying Figure 4.7A by making 0 recede far away to the left, we see how a translation arises as the limiting form of a central dilatation A'B' = kAB when k tends to 1. Still more easily, we can change Figure 4.7B so as to make 0 the midpoint of AA'; thus the central dilatation A'B' = -kAB includes, as a special case, the half-turn A'B' = - AB, %3D for which ABA'B' is a parallelogram with center 0. Figure 4.7B EXERCISES 1. What is the locus of the midpoint of a segment of varying length such that one end remains fixed while the other end runs around a circle? 2. Given an acute-angled triangle ABC, construct a square with one side lying on BC while the other two vertices lie on CA and AB, respectively. 4.8 Spiral similarity If a figure is first dilated and then translated, the final figure and the original figure still have corresponding lines parallel, so that the result is simply a dilatation. More generally, and for the same of any two dilatations (i.e. the effect of first performing one, then the other dilatation) is a dilatation. On the other hand, if a figure is first dilated and then rotated, corresponding lines are no longer parallel. Thus the sum of a dilatation and a rotation (other than the identity or a half-turn) is not a dilatation, though it is still a direct similarity, preserving angles in both magnitude and sign. Lson, the sum 96 TRANSFORMATIONS The sum of a central dilatation and a rotation about the same center is called a "dilative rotation" or spiral similarity. This little known transformation can be used in the solution of many problems. If, as in Figure 4.8A, a spiral similarity with center O takes AB to A'B', then AOAB and A0A'B' are directly similar, and ZAQA' = LBOB'. Moreover, as in the tase ura sumpie unataivi, the ratio of magnifica-
Transcribed Image Text:9:46 1 LTE A aproged.pt OA' = - kOA OB' – kOB, and DILATATION 95 as in Figure 4.7B. Remembering that parallel lines cut transversals into proportional segments, we easily deduce that C' lies on OC; in fact, Varying Figure 4.7A by making 0 recede far away to the left, we see how a translation arises as the limiting form of a central dilatation A'B' = kAB when k tends to 1. Still more easily, we can change Figure 4.7B so as to make 0 the midpoint of AA'; thus the central dilatation A'B' = -kAB includes, as a special case, the half-turn A'B' = - AB, %3D for which ABA'B' is a parallelogram with center 0. Figure 4.7B EXERCISES 1. What is the locus of the midpoint of a segment of varying length such that one end remains fixed while the other end runs around a circle? 2. Given an acute-angled triangle ABC, construct a square with one side lying on BC while the other two vertices lie on CA and AB, respectively. 4.8 Spiral similarity If a figure is first dilated and then translated, the final figure and the original figure still have corresponding lines parallel, so that the result is simply a dilatation. More generally, and for the same of any two dilatations (i.e. the effect of first performing one, then the other dilatation) is a dilatation. On the other hand, if a figure is first dilated and then rotated, corresponding lines are no longer parallel. Thus the sum of a dilatation and a rotation (other than the identity or a half-turn) is not a dilatation, though it is still a direct similarity, preserving angles in both magnitude and sign. Lson, the sum 96 TRANSFORMATIONS The sum of a central dilatation and a rotation about the same center is called a "dilative rotation" or spiral similarity. This little known transformation can be used in the solution of many problems. If, as in Figure 4.8A, a spiral similarity with center O takes AB to A'B', then AOAB and A0A'B' are directly similar, and ZAQA' = LBOB'. Moreover, as in the tase ura sumpie unataivi, the ratio of magnifica-
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