A triangle with vertices (4, 0), (-1, -1) and (3, 5) is:

Concept:

Distance between two points:

The distance between any two points \(\rm (x_1, y_1)\) and \(\rm(x_2, y_2)\) is given by the distance formula as follows: \(d = \sqrt {(x_2 - x_1)^2 + (y_2 - y_1)^2}\)

Inverse of the Pythagorean theorem:

If a triangle with sides a, b and c satisfies the equation \(\rm c^2=a^2+b^2\) where is the longest side, then the triangle is always right angled triangle.

Calculation:

First we will calculate the distance between all pairs of points from the given pairs.

First consider the points (4, 0) and (-1, -1).

The distance is given as follows:

\(\rm d_1 = \sqrt{(4-(-1))^2+(0-(-1))^2}\\ = \sqrt{25+1}\\ =\sqrt{26} \)

Now consider the pair (-1, -1) and (3, 5).

The distance is given as follows:

\(\rm d_2 = \sqrt{(3-(-1))^2+(5-(-1))^2}\\= \sqrt{16+36}\\= 2\sqrt {13}\)

Now consider the pair (4, 0) and (3, 5).

The distance is given as follows:

\(\begin{align*} d_3 &= \sqrt{(3-(4))^2+(5-(0))^2}\\ &= \sqrt{1+25}\\ &=\sqrt{26} \end{align*}\)

Therefore, two distances are equal thus, the triangle is isosceles.

The greatest side is of length \(2\sqrt{13}\).

As we can see that, d₂² = d₁² + d₃²

Therefore, by using inverse of the Pythagorean theorem, we conclude that the triangle is right-angled.

Hence, option 1 is correct