Explanation:
To determine the type of triangle formed by the three roads, we need to use the Triangle Inequality Theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Let's check if this theorem holds true for the three sides:
- 15 + 18 > 20 (True)
- 15 + 20 > 18 (True)
- 18 + 20 > 15 (True)
Since the sum of the lengths of any two sides is greater than the length of the third side, we can conclude that a triangle can be formed.
To determine the type of triangle, we can use the Pythagorean Theorem, which states that in a right triangle, the sum of the squares of the two shorter sides equals the square of the longest side.
Let's check if this theorem holds true for the three sides:
- 15² + 18² = 225 + 324 = 549
- 20² = 400
Since 15² + 18² is less than 20², we can conclude that the triangle is not a right triangle.
Next, we can use the Law of Cosines to determine the type of triangle. The Law of Cosines states that in a triangle with side lengths a, b, and c, and angles A, B, and C opposite those sides, the following equation holds true:
c² = a² + b² - 2ab cos(C)
Let's calculate the cosine of each angle:
- cos(A) = (18² + 20² - 15²) / (2 * 18 * 20) = 0.425
- cos(B) = (15² + 20² - 18²) / (2 * 15 * 20) = 0.3
- cos(C) = (15² + 18² - 20²) / (2 * 15 * 18) = 0.208
Using these values, we can calculate the value of c² for each angle:
- c² = 18² + 20² - 2(18)(20)(0.425) = 344.4
- c² = 15² + 20² - 2(15)(20)(0.3) = 331
- c² = 15² + 18² - 2(15)(18)(0.208) = 304.68
Since none of these values equals the sum of the squares of the other two sides, we can conclude that the triangle is not a right triangle either.
To determine the type of triangle, we can look at the values of the cosine of each angle. If the cosine of an angle is greater than 0, the angle is acute. If the cosine of an angle is less than 0, the angle is obtuse. If the cosine of an angle is equal to 0, the angle is a right angle.
From the values we calculated above, we can see that the cosine of each angle is positive, which means that all angles are acute. Therefore, the answer is:
D: acute triangle.