To determine the type of triangle formed by the three cities A, B, and C, we need to use the triangle inequality theorem and the properties of triangles to evaluate the sum of the squares of the sides. Here are the given distances:
- [tex]\( AB = 22 \)[/tex] miles
- [tex]\( BC = 54 \)[/tex] miles
- [tex]\( AC = 51 \)[/tex] miles
First, let's calculate the squares of each side:
- [tex]\( AB^2 = 22^2 = 484 \)[/tex]
- [tex]\( BC^2 = 54^2 = 2916 \)[/tex]
- [tex]\( AC^2 = 51^2 = 2601 \)[/tex]
To determine the type of triangle, we use the following observations:
1. For an acute triangle:
- The sum of the squares of any two sides should be greater than the square of the third side.
2. For an obtuse triangle:
- The sum of the squares of any two sides should be less than the square of the third side for one specific pair.
Now, let's check each condition:
- Calculate [tex]\( AB^2 + AC^2 \)[/tex] versus [tex]\( BC^2 \)[/tex]:
[tex]\[ 484 + 2601 = 3085 \][/tex]
- Since [tex]\( 3085 > 2916 \)[/tex], this satisfies the condition for an acute triangle.
- Therefore, the statement "an acute triangle, because [tex]\( 22^2 + 51^2 > 54^2 \)[/tex]" is correct.
- Calculate [tex]\( AB^2 + BC^2 \)[/tex] versus [tex]\( AC^2 \)[/tex]:
[tex]\[ 484 + 2916 = 3400 \][/tex]
- Since [tex]\( 3400 > 2601 \)[/tex], this also satisfies the condition for an acute triangle.
- Therefore, the statement "an acute triangle, because [tex]\( 22^2 + 54^2 > 51^2 \)[/tex]" is also correct.
Thus, both conditions confirm that the triangle is an acute triangle. Therefore, the correct statements are:
- An acute triangle, because [tex]\( 22^2 + 51^2 > 54^2 \)[/tex]
- An acute triangle, because [tex]\( 22^2 + 54^2 > 51^2 \)[/tex]
This verifies that the triangle created by the three cities is an acute triangle.
