Geometry: Common Core (15th Edition)

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

Chapter 8 - Right Triangles and Trigonometry - Common Core Cumulative Standards Review - Selected Response - Page 540: 18

Answer

a. $x = 7$ b. Diagonals $AC$ and $DB$ are $9$.

Work Step by Step

a. The diagram is that of an isosceles trapezoid; therefore, the diagonals are congruent to one another. The diagonals of this trapezoid are $\overline{DB}$ and $\overline{AC}$. Set the two diagonals equal to one another: $\overline{DB}$ = $\overline{AC}$ Plug in what we know: $(2x - 8) + (x - 4) = x + 2$ Rewrite the equation without parentheses: $2x - 8 + x - 4 = x + 2$ Combine like terms on the left side of the equation: $3x - 12 = x + 2$ Subtract $x$ from both sides of the equation to move variables to the left side of the equation: $2x - 12 = 2$ Add $12$ to both sides of the equation to move constants to the right side of the equation: $2x = 14$ Divide each side by $2$ to solve for $x$: $x = 7$ b. Let's plug in $x$ into the expression for $\overline{AC}$ to find the length of this diagonal: $AC = 7 + 2$ Add to solve: $AC = 9$ Since the two diagonals are congruent to each other, $DB$ is also $9$.
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