Explanation:
To determine the type of triangle formed by the points \( A(1, 5) \), \( B(4, 7) \), and \( C(5, 3) \), we first need to calculate the lengths of the sides of the triangle and then use these lengths to classify the triangle.

Step-by-Step Solution:

1. Calculate the lengths of the sides using the distance formula:

The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

- Length \( AB \):
\[
AB = \sqrt{(4 - 1)^2 + (7 - 5)^2} = \sqrt{3^2 + 2^2} = \sqrt{9 + 4} = \sqrt{13}
\]

- Length \( BC \):
\[
BC = \sqrt{(5 - 4)^2 + (3 - 7)^2} = \sqrt{1^2 + (-4)^2} = \sqrt{1 + 16} = \sqrt{17}
\]

- Length \( CA \):
\[
CA = \sqrt{(5 - 1)^2 + (3 - 5)^2} = \sqrt{4^2 + (-2)^2} = \sqrt{16 + 4} = \sqrt{20}
\]

2. Check if the triangle is right-angled using the Pythagorean theorem:

For a triangle to be right-angled, the square of the longest side must be equal to the sum of the squares of the other two sides.

- Compute \( \sqrt{20} \) as \( CA \) is the longest side:
\[
\sqrt{20}^2 = 20
\]
- Check if:
\[
(\sqrt{13})^2 + (\sqrt{17})^2 = 13 + 17 = 30
\]
\[
20 \neq 30 \quad (\text{Thus, not a right-angle triangle})
\]

3. Determine the type of triangle:

- Scalene: A triangle where all three sides have different lengths.
- Here, \( \sqrt{13} \neq \sqrt{17} \neq \sqrt{20} \), so the triangle is scalene.

- Isosceles: A triangle with at least two equal sides.
- Since all three sides are different, it is not isosceles.

- Equilateral: A triangle with all three sides equal.
- Clearly, the sides are not all equal.


The triangle \( ABC \) is Scalene with no right angle.

The correct answer is B. Scalene with no right angle