A triangular plot of land has one side along a straight road measuring 336 feet. A second side makes a \( 39^{\circ} \) angle with the road, and the third How long are the other two sides? The longer side of the triangular plot is \( \square \) feet. The shorter side of the triangular plot is \( \square \) feet. (Round to the nearest hundredth as needed.)

To solve this problem, we can use the Law of Sines. Let's denote the sides of the triangle as follows: - Let \( a = 336 \) feet (the side along the road). - Let \( B = 39^\circ \) (the angle opposite side \( a \)). - Let \( C \) be the angle opposite the longer side, which we will denote as \( b \). - Let \( A \) be the angle opposite the shorter side, which we will denote as \( c \). Since we don't have the angles \( A \) and \( C \), we need to find them first. We can use the fact that the sum of angles in a triangle is \( 180^\circ \): \[ A + B + C = 180^\circ \] We can express \( C \) in terms of \( A \): \[ C = 180^\circ - A - 39^\circ = 141^\circ - A \] Next, we can use the Law of Sines: \[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \] From this, we can express \( b \) and \( c \): \[ b = \frac{a \cdot \sin B}{\sin A} = \frac{336 \cdot \sin(39^\circ)}{\sin A} \] \[ c = \frac{a \cdot \sin C}{\sin A} = \frac{336 \cdot \sin(141^\circ - A)}{\sin A} \] To find the lengths of sides \( b \) and \( c \), we need to know the angle \( A \). However, without additional information about the triangle, we cannot determine \( A \) directly. If we assume that the triangle is a right triangle, we can set \( A = 90^\circ \). In that case: \[ C = 141^\circ - 90^\circ = 51^\circ \] Now we can calculate the lengths of the other two sides using the Law of Sines: 1. Calculate \( b \): \[ b = \frac{336 \cdot \sin(39^\circ)}{\sin(90^\circ)} = 336 \cdot \sin(39^\circ) \] Calculating \( \sin(39^\circ) \): \[ \sin(39^\circ) \approx 0.6293 \] So, \[ b \approx 336 \cdot 0.6293 \approx 211.00 \text{ feet} \] 2. Calculate \( c \): \[ c = \frac{336 \cdot \sin(51^\circ)}{\sin(90^\circ)} = 336 \cdot \sin(51^\circ) \] Calculating \( \sin(51^\circ) \): \[ \sin(51^\circ) \approx 0.7771 \] So, \[ c \approx 336 \cdot 0.7771 \approx 261.00 \text{ feet} \] Thus, the lengths of the sides are: - The longer side \( b \) is approximately \( 261.00 \) feet. - The shorter side \( c \) is approximately \( 211.00 \) feet. Final answers: - The longer side of the triangular plot is \( 261.00 \) feet. - The shorter side of the triangular plot is \( 211.00 \) feet.