Problem 18 - Algebra 2 and Trigonometry

Question

Ross said that when he jogs, his path forms a triangle. Two sides of the triangle are 2.0 kilo- meters and 2.5 kilometers in length and the angle opposite the shorter side measures 45 degrees. Rosa said that when she jogs, her path also forms a triangle with two sides of length 2.0 kilometers and 2.5 kilometers and an angle of 45 degrees opposite the shorter side. Rosa said that her route is longer than the route Ross follows. Is this possible? Explain your answer.

Step-by-Step Solution

Answer

Rosa's route cannot be longer than Ross's; their paths are identical.

Step 1: Understand Both Scenarios

Ross and Rosa both jog triangles with the same side lengths of 2.0 km and 2.5 km, and both have an angle of 45 degrees opposite the shorter side of 2.0 km.

Step 2: Use the Law of Cosines for Ross

Apply the law of cosines to find the third side \(c\) of Ross's triangle: \[c^2 = 2.0^2 + 2.5^2 - 2 \cdot 2.0 \cdot 2.5 \cdot \cos(45^{\circ})\]. Compute \(c\): \[c^2 = 4 + 6.25 - 5 \cdot \frac{\sqrt{2}}{2}\]. Simplifying, \[c^2 = 10.25 - 3.5355\], thus \[c^2 = 6.7145\], leading to \(c \approx 2.59\) km.

Step 3: Use the Law of Cosines for Rosa

Similarly, find the third side \(c'\) for Rosa using the law of cosines: \[c'^2 = 2.0^2 + 2.5^2 - 2 \cdot 2.0 \cdot 2.5 \cdot \cos(45^{\circ})\]. The calculation will be the same as Ross's, resulting in \(c' = c \approx 2.59\) km.

Step 4: Calculate Perimeter for Both Routes

For both, add all the sides to get the perimeter of each triangle. Ross's perimeter is \(2.0 + 2.5 + 2.59 = 7.09\) km. Rosa's is also \(2.0 + 2.5 + 2.59 = 7.09\) km.

Step 5: Conclude and Explain

Both Ross and Rosa have triangles with the same side lengths, so their paths (perimeters) are identical. Rosa cannot have a longer route than Ross.

Key Concepts

Triangle Geometry

Triangles are fundamental shapes in geometry, characterized by three sides, three angles, and three vertices connecting these sides. In the scenario where both Ross and Rosa jog along paths forming two such triangles, the inherent symmetry of triangles becomes apparent. All triangles possess interior angles that sum up to 180 degrees, and these angles influence the lengths of the sides.

Ross's and Rosa's jogging paths can be modeled as triangles with two known side lengths: one 2.0 km long and the other 2.5 km. The third side and the other angles can be deduced using trigonometric principles like the Law of Cosines. Understanding the relations between sides and angles is key to confirming that the triangles have identical dimensions and properties, leading to the same perimeter measurement.

Perimeter Calculation

The perimeter of a triangle is the total distance around the triangle, calculated by adding up the lengths of its three sides. For Ross and Rosa's jogging paths, this means summing the two known side lengths with the third calculated side.

Using the Law of Cosines, we find that both Ross's and Rosa's third side measures approximately 2.59 km. Therefore, their perimeters are computed as follows: 2.0 km + 2.5 km + 2.59 km, which equals 7.09 km.

Perimeter calculation is straightforward once all the sides of a triangle are known and is a critical step in comparing the lengths of their jogging routes.

Trigonometry

Trigonometry is the study of the relationships between the angles and sides of triangles. It provides the tools to solve for unknown sides and angles based on known data, especially when an angle and two sides are given, as with Ross and Rosa.

The Law of Cosines is one such tool in trigonometry, given as: \[c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\]where \(a\), \(b\), and \(c\) are the sides of a triangle, with \(C\) being the angle opposite side \(c\). This formula allows us to calculate the unknown third side when two sides and the included angle are known.

Applying this to Ross and Rosa’s triangles, we see that despite having fixed side lengths and an angle, the third side's calculation remains consistent for both, affirming the use of trigonometric techniques in their comparison.

Angle Measurement

Angles are crucial in defining a triangle's shape and determine which trigonometric principles apply. Ross and Rosa are both dealing with an angle of 45 degrees opposite the 2.0 km side. This angle helps calculate the third side length using trigonometric principles.

Measuring an angle precisely is vital as even slight variations can lead to different outcomes in geometrical calculations. In trigonometry, angle measurement works in degrees or radians, where a full circle is 360 degrees or \(2\pi\) radians.

For Ross and Rosa, the 45-degree angle ensures that their triangles are identical in configuration, confirming that their respective routes share the same perimeter. Accurately measuring this initial angle guarantees consistency in calculating other triangle properties.