To determine the type of triangle based on the lengths of its sides, we can use the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
Let's denote the sides as follows:
- \( a = 56 \mathrm{~km} \)
- \( b = 8.5 \mathrm{~km} \)
- \( c = 56.5 \mathrm{~km} \)
We are assuming that \( c \) is the longest side, which would be the hypotenuse if the triangle is right-angled.
According to the Pythagorean theorem:
\( a^2 + b^2 = c^2 \) for a right-angled triangle.
Let's calculate \( a^2 + b^2 \) and compare it to \( c^2 \) to see if we have a right-angled triangle.
\( a^2 = 56^2 = 3136 \mathrm{~km}^2 \)
\( b^2 = 8.5^2 = 72.25 \mathrm{~km}^2 \)
\( a^2 + b^2 = 3136 \mathrm{~km}^2 + 72.25 \mathrm{~km}^2 = 3208.25 \mathrm{~km}^2 \)
Now, let's calculate \( c^2 \):
\( c^2 = 56.5^2 = 3192.25 \mathrm{~km}^2 \)
Comparing \( a^2 + b^2 \) with \( c^2 \):
\( 3208.25 \mathrm{~km}^2 \) (sum of squares of the shorter sides) is greater than \( 3192.25 \mathrm{~km}^2 \) (square of the longest side).
Since \( a^2 + b^2 > c^2 \), the triangle is not right-angled. Instead, it is an acute triangle because the sum of the squares of the two shorter sides is greater than the square of the longest side, indicating that all angles in the triangle are less than 90 degrees.
**The accurate answer is: Acute.**
