Square metric not satisfying the SAS postulate

[pholee95]

I'm not sure on how to do this problem. Can someone please help and explain? Thank you!

Recall (Exercise 3.2.8) that the square metric distance between two points (x1, y1) and
(x2, y2) in R^2 is given by D((x1, y1), (x2, y2))= max{|x2 − x1|, |y2 − y1|}. Show by
example that R^2 with the square metric and the usual angle measurement function does
not satisfy the SAS Postulate.

 

[I like Serena]

I'm not sure on how to do this problem. Can someone please help and explain? Thank you!

Recall (Exercise 3.2.8) that the square metric distance between two points (x1, y1) and
(x2, y2) in R^2 is given by D((x1, y1), (x2, y2))= max{|x2 − x1|, |y2 − y1|}. Show by
example that R^2 with the square metric and the usual angle measurement function does
not satisfy the SAS Postulate.

Hi pholee95! Welcome to MHB! ;)

Let's pick a simple triangle, like a rectangular one with points (0,0), (1,0), (0,1).
Now let's rotate it by 45 degrees, keeping the angles the same, and keeping the lengths of the side according to the metric the same.
Then we'll get the triangle with points (0,0), (1,1), (1,-1).
According to the SAS postulate it should then be congruent.
But congruency requires that the lengths of all sides are the same. Is the (metric) length of the 3rd side the same?

 

[pholee95]

Hi pholee95! Welcome to MHB! ;)

Let's pick a simple triangle, like a rectangular one with points (0,0), (1,0), (0,1).
Now let's rotate it by 45 degrees, keeping the angles the same, and keeping the lengths of the side according to the metric the same.
Then we'll get the triangle with points (0,0), (1,1), (1,-1).
According to the SAS postulate it should then be congruent.
But congruency requires that the lengths of all sides are the same. Is the (metric) length of the 3rd side the same?

No it won't be the same. Right?

 

[I like Serena]

No it won't be the same. Right?

Correct. All sides of the first triangle have metric length 1.
But the second triangle has metric lengths 1, 1, and 2.