Elementary Geometry for College Students (5th Edition)

Published by Brooks Cole
ISBN 10: 1439047901
ISBN 13: 978-1-43904-790-3

Chapter 10 - Section 10.3 - Preparing to Do Analytic Proofs - Exercises - Page 473: 16

Answer

R(0,0) S(s,0) V(0,t) T(s,t)

Work Step by Step

First, we find the coordinates of R. From the picture, we see vertex R is located at the origin. Hence the coordinates of R are $(0,0)$. Next, we find the coordinates of S. We are given the $x$-coordinate of S, which is $s$, and from the picture, we see S lies on the $x$-axis, which means the $y$-coordinate of S is $0$. Thus vertex S has coordinates $(s,0).$ Thirdly, we find the coordinates of vertex V. We are given that the $y$-coordinate of V is $t$, and we see from the picture that V lies on the $y$-axis. Hence the $x$-coordinate of V is zero, which means the coordinates of V are $(0,t).$ Lastly, we find the coordinates of vertex T. We are given that figure RSTV is a rectangle. So $\overline{ST}$ is parallel to $\overline{RV}$ and their lengths are equal. Thus since $\overline{RV}$ lies along the $y$-axis, it is a vertical line segment. This means $\overline{ST}$ is also a vertical line segment, and since V lies $t$ units above R, T must lie t units above S. Hence the $y$-coordinate of T is $t$. Similarly, $\overline{VT}$ is parallel to the the horizontal line segment $\overline{RS}$ and S lies $s$ units to the right of R. So T must lie $s$ units to the right of V. Hence the $x$-coordinate of T is $s$, which means the coordinates of T are $(s,t).$
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