Dallas, Austin, and Houston, Texas form a triangle.
The angle at Dallas measures 43°.
The angle at Austin measures 83°.
The angle at Houston measures 54°.
Between which two cities is the distance the greatest?
Answer (2)
Using the Law of Sines, we find that the distance between Austin and Dallas is the greatest due to the comparison of the sine values of their angles. Since the angle opposite the side connecting Austin is the largest, this distance is the longest. Therefore, Austin and Dallas have the greatest distance among the three cities.
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To solve this problem, we need to determine which side of the triangle formed by Dallas, Austin, and Houston is the longest. We are given the angles at each city: Dallas at 43°, Austin at 83°, and Houston at 54°.
In a triangle, the longest side is opposite the largest angle. Let's first identify the angles:
Dallas: 43°
Austin: 83°
Houston: 54°
Among these angles, 83° is the largest angle. Therefore, the longest side of the triangle will be opposite the angle at Austin.
This means the greatest distance is between the cities located at Dallas and Houston since the angle at Austin, which is 83°, lies opposite this side.
So, the distance between Dallas and Houston is the greatest.
