Question

Compute and compare the side lengths of the two figures. Does this support an argument claiming that they are congruent? What else must be true to say two polygons are congruent?

Answer 1

We compute for the side lengths using the distance formula √[(x₂-x₁)²+(y₂-y₁)²].

AB = √[(-7--5)²+(4-7)²] = √13
A'B' = √[(-9--7)²+(0-3)²] = √13

BC = √[(-5--3)²+(7-4)²] = √13
B'C' = √[(-7--5)²+(3-0)²] =√13

CD = √[(-3--5)²+(4-1)²] = √13
C'D' = √[(-5--7)²+(0--3)²] = √13

DA = √[(-5--7)²+(1-4)²] = √13
D'A' = √[(-7--9)²+(-3-0)²] = √13

The two polygons are squares with the same side lengths. 

But this is not enough information to support the argument that the two figures are congruent. In order for the two to be congruent, they must satisfy all conditions: 
1. They have the same number of sides.
2. All the corresponding sides have equal length.
3. All the corresponding interior angles have the same measurements.

The third condition was not proven.

Answer 2

The side lengths of the two figures can be calculated with the distance formula:

 √ (x1 - x2)² + (y1 - y2)²        

AB = √13  

BC = √13      

CD = √13    

DA = √13  

A'B' = √13      

B'C' = √13      

C'D' = √13      

D'A' = √13

They are all the same lengths. This supports an argument claiming that they are congruent because if the sides are congruent, then the two polygons are congruent.

Something else that must be true to say that two polygons are congruent is that the angles have equal measures.

Hope this helps! :)