Answer:

Given points A(-4, 0), B(1, -4), and C(5, 1) are the vertices of a triangle. To determine the type of triangle formed by these points, we need to analyze the lengths of the sides of the triangle.

Step 1: Calculate the lengths of the sides of the triangle
To calculate the lengths of the sides of the triangle, we can use the distance formula:

Distance = √((x2 - x1)^2 + (y2 - y1)^2)

Using the distance formula, we can calculate the lengths of the sides AB, BC, and AC:

AB = √((1 - (-4))^2 + (-4 - 0)^2) = √(5^2 + 4^2) = √(25 + 16) = √41
BC = √((5 - 1)^2 + (1 - (-4))^2) = √(4^2 + 5^2) = √(16 + 25) = √41
AC = √((5 - (-4))^2 + (1 - 0)^2) = √(9^2 + 1^2) = √(81 + 1) = √82

Step 2: Determine the type of triangle
Now that we have the lengths of the sides AB, BC, and AC, we can analyze them to determine the type of triangle.

An isosceles right-angled triangle has two sides of equal length and one right angle (90 degrees). In this case, the length of AB is equal to the length of BC, which is √41. However, the length of AC is √82, which is not equal to √41. Therefore, the triangle formed by the given points A, B, and C is not an isosceles right-angled triangle.

An equilateral triangle has all three sides of equal length. In this case, the lengths of AB, BC, and AC are not equal. Therefore, the triangle formed by the given points A, B, and C is not an equilateral triangle.

A scalene triangle has all three sides of different lengths. In this case, the lengths of AB and BC are equal (√41), but the length of AC (√82) is different. Therefore, the triangle formed by the given points A, B, and C is not a scalene triangle.

Therefore, the correct answer is option 'A': None of these. The triangle formed by the given points A(-4, 0), B(1, -4), and C(5, 1) is none of the types mentioned above.