The points A(0, 6), B(-5, 3), and C(3, 1) are the vertices of a triangle, this triangle is-

Concept -

(i) The distance formula is given by:

\( d = √{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)

(ii)  If all three sides are of different lengths, the triangle is a scalene triangle.
(iii) If two sides are of equal length, the triangle is an isosceles triangle.
(iv)  If all three sides are of equal length, the triangle is an equilateral triangle.

Explanation -

1. Length of side AB:

\(AB = √{(-5 - 0)^2 + (3 - 6)^2} = √{25 + 9} = √{34} \)

2. Length of side BC:

\(BC = √{(3 - (-5))^2 + (1 - 3)^2} = √{64 + 4} = √{68} \)

3. Length of side CA:

\(CA = √{(0 - 3)^2 + (6 - 1)^2} = √{9 + 25} = √{34} \)

In this case, AB and CA have the same length √34, while BC has a different length √68 .

here  \(AB^2 + CA^2 = BC^2 \)  (Since the triangle satisfies the Pythagorean theorem, meaning it is a right-angled triangle) .

Therefore, the triangle ABC is an Isosceles Right-Angled Triangle.

Hence Option 3 is the correct answer.