Do the points make a right triangle? (4,0) (2,1) (-1,-5)?

Method 1 :

A(4, 0), B(2, 1) and C(-1, -5)
Slope of AB = (0 - 1) / (4 - 2) = -1/2
Slope of BC = (-5 - 1) / (-1 - 2) = 2

(Slope of AB) * (Slope of BC) = (-1/2) * 2 = -1
Hence, ∠ABC = 90°
Thus ABC is a right triangle


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Method 2 :

A(4, 0), B(2, 1) and C(-1, -5)
AB = √[(4 - 2)² - (0 - 1)²] = √5
BC = √[(-1 - 2)² - (-5 - 1)²] = √45
AC = √[(-1 - 4)² - (-5 - 0)²] = √50

AB² + BC² = (√5)² + (√45)² = 50
AC² = (√50)² = 50

AC² = AB² + BC²
By the reverse theorem of Pythagoras theorem, ABC is a right triangle where ∠ABC = 90°

Pythagoras
sIde lengths
50 = 5 + 45
Yes it is a right triangle

as long as they don't all land on the same line, then yes.

So to determine this, let's get the slope of the line connecting the first two points and the first/last points. If the slopes are the same then this is not a triangle.

Slope is change in y over change in x, so:

m = (0 - 1) / (4 - 2) and m = (0 - (-5)) / (4 - (-1))
m = -1 / 2 and m = (0 + 5) / (4 + 1)
m = -1 / 2 and m = 5 / 5
m = -1 / 2 and m = 1

So the slopes are different when coming out of the same common point, so they do make a triangle.