what is the other way?

Question

Show in two ways that the points (6,1) (2,-3) and (4,-5) are the verices of a right triangle.


the 1st way is the pythagorean theorem...

what is the other way?

Answer 1

The easiest way that occurs to me is to show that two segments are perpendicular. This happens when their slopes are negative reciprocals of each other (m1 = -1/m2). If that's the case, it means the two segments meet at a right angle, which makes the triangle a right triangle.

Using the slope formula (y2-y1)/(x2-x1):

(6,1) to (2,-3) -> (-3-1)/(2-6) = -4/-4 = 1
(4,-5) to (6,1) -> (1-(-5))/(6-4) = 6/2 = 3
(2,-3) to (4,-5) -> (-5-(-3))/(4-2) = -2/2 = -1

The slope from (6,1) to (2,-3) (1) is the negative reciprocal of the slope from (2,-3) to (4,-5) (-1). (-1/m1 =? m2 -> -1/1 =? -1 -> -1 = -1). This means that (2,-3) is the vertex that is a right angle.

Answer 2

Well if this is for homework i noe that there are many many many ways to figure it out.
You can compare the size of the two points.
If it 5 boxes. The area if 5*5 which is 25.
So u add up A and B and get C.
That is also another way to do it.
also you can draw a diagram to show urself.

Answer 3

let A (6,1) , B(2,-3) and C(4,-5)

vector AB=<-4,-4>
vector AC=<-2,-6>
vector BC=<2,-2>

dot product of vectors AB and BC we get 0 so it means they are perpendicular so the triangle ABC is right one.