The points (0, 83), (1, 3), and (82, 30) form an obtuse angled triangle, lie on a straight line, form an acute angled triangle, or form a right angled triangle?

Given (0, 83), (1, 3), and (82, 30)
Let A = (0, 83), B = (1, 3), C = (82, 30)
Slope of AB = (3 - 83) / (1 - 0) = -80
Slope of BC = (30 - 3) / (82 - 1) = 27 / 81 = 1/3
Since the slopes are different, the points do not lie on a straight line.
Now, let's find the lengths of the sides using the distance formula:
AB = sqrt((1-0)^2 + (3-83)^2) = sqrt(1 + 6400) = sqrt(6401)
BC = sqrt((82-1)^2 + (30-3)^2) = sqrt(81^2 + 27^2) = sqrt(6561 + 729) = sqrt(7290)
AC = sqrt((82-0)^2 + (30-83)^2) = sqrt(82^2 + (-53)^2) = sqrt(6724 + 2809) = sqrt(9533)
Let's check if it's a right-angled triangle using the Pythagorean theorem:
AB^2 + BC^2 = 6401 + 7290 = 13691
AC^2 = 9533
Since AB^2 + BC^2 ≠ AC^2, it's not a right-angled triangle.
Now let's check if it's an obtuse or acute angled triangle.
If AB^2 + BC^2 > AC^2, then it's an acute angled triangle.
If AB^2 + BC^2 < AC^2, then it's an obtuse angled triangle.
13691 > 9533
Therefore, it is an acute angled triangle.