Geometry: Common Core (15th Edition)

Published by Prentice Hall
ISBN 10: 0133281159
ISBN 13: 978-0-13328-115-6

Chapter 7 - Similarity - 7-4 Similarity in Right Triangles - Practice and Problem-Solving Exercises - Page 467: 53


$x = 3$ $y = 4$

Work Step by Step

Parallelograms have opposite sides that are congruent. In order for $RSTV$ to be a parallelogram, we must have $RV$ and $TS$ equal to one another and $VT$ and $SR$ equal to one another. Let's set $RV$ and $TS$ equal to one another first: $RV = TS$ Let's plug in what we are given in the diagram: $2x + 3 = y + 5$ Now, let's set $VT$ and $SR$ equal to one another: $VT = SR$ Plug in what we are given: $5x = 4y - 1$ We have two equations and two variables. We can use the elimination method by setting up the system of equations to solve for one variable: $2x + 3 = y + 5$ $5x = 4y - 1$ Let's get all the variables on one side and the constants on the other: $2x - y = 2$ $5x - 4y = -1$ We have to modify one of the equations because we need one of the variables in both equations to differ only in sign. Let's multiply the first equation by $-4$: $-8x + 4y = -8$ $5x - 4y = -1$ Now, we can add the two equations together: $-3x = -9$ Divide each side by $-3$ to solve for $x$: $x = 3$ Now that we have the value for $x$, we can plug in this value for $x$ into one of the original equations to find $y$: $5x = 4y - 1$ Substitute $3$ for $x$: $5(3) = 4y - 1$ Multiply to simplify: $15 = 4y - 1$ Add $1$ to both sides of the equation to solve for $y$: $4y = 16$ Divide each side by $4$ to solve for $y$: $y = 4$
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