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:
$$(\sqrt{13}{)}^2+(\sqrt{17}{)}^2=13+17=30$$
$$20\ne 30(\text{Thus, not a right-angle triangle})$$

3. Determine the type of triangle:

- Scalene: A triangle where all three sides have different lengths.
- Here, $\sqrt{13}\ne \sqrt{17}\ne \sqrt{20}$, 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