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 538: 1



Work Step by Step

The area of a rectangle is given by the formula $A = lw$, where $l$ is the length and $w$ is the width of the rectangle. We are only given the width in this exercise. We need to find the length. In the diagram, a diagonal is drawn such that the rectangle is divided into two right triangles. We know that the sides of a triangle have specific relationships. Since we also have the measure of an acute angle, we can use one of the trigonometric ratios to find the length of the rectangle. We are given the measurement of the side opposite to the angle. We want to find the side that is adjacent to the angle, which would be the length of the rectangle. Let's use the tangent ratio, which is given as: tan θ = $\frac{opposite}{adjacent}$ Let's plug in what we know: tan $32^{\circ} = \frac{8}{x}$ Multiply each side by $x$: tan $32^{\circ}(x) = 8$ Divide each side of the equation by tan $32^{\circ}$ to solve for $x$: $x \approx 12.8$ cm Now, we have both the length and width of the rectangle, so we can calculate its area: $A$ \approx ($12.8$ cm)($8$ cm) Multiply to solve: $A \approx 102.4$ cm$^2$ The answer is option $A$.
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