Square metric not satisfying the SAS postulate

In summary: Correct. All sides of the first triangle have metric length 1.According to the SAS postulate, the triangles should be congruent.According to the SAS postulate, the triangles should be congruent.
  • #1
pholee95
10
0
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.
 
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  • #2
pholee95 said:
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?
 
  • #3
I like Serena said:
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?
 
  • #4
pholee95 said:
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.
 

1. What does it mean for a square to not satisfy the SAS postulate?

The SAS (Side-Angle-Side) postulate is a rule in geometry that states if two triangles have two sides and the included angle of one triangle equal to two sides and the included angle of the other triangle, then the triangles are congruent. If a square does not satisfy this postulate, it means that the sides and angles of the square do not match up with another triangle, making it impossible for the two shapes to be congruent.

2. Why is it important for squares to satisfy the SAS postulate?

The SAS postulate is important because it is one of the ways to prove that two triangles are congruent. If a square does not satisfy this postulate, it means that it cannot be proven to be congruent to any other triangle using this method. This could lead to incorrect conclusions and misunderstandings in geometry.

3. Can a square ever not satisfy the SAS postulate?

Yes, a square can fail to satisfy the SAS postulate. This can happen if the given side lengths and angles of the square do not match up with any other triangle. For example, a square with side lengths of 4cm and an included angle of 90 degrees cannot be proven to be congruent to any other triangle using the SAS postulate.

4. What other postulates or theorems can be used to prove the congruency of squares?

There are several other postulates and theorems that can be used to prove the congruency of squares. These include the SSS (Side-Side-Side) postulate, the ASA (Angle-Side-Angle) theorem, and the HL (Hypotenuse-Leg) theorem. These methods rely on different combinations of known side lengths and angles to prove congruency.

5. Are there any real-world implications of a square not satisfying the SAS postulate?

In most cases, the SAS postulate and the congruency of squares are not relevant in real-world applications. However, in some cases, such as in architecture or engineering, it may be important to accurately determine the congruency of shapes for building structures or designing objects. In these cases, the SAS postulate and other methods of proving congruency may be used to ensure accuracy and avoid errors.

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