Use the distance formula to compute the distance from A(0,0) to B(1,3) and you should get sqrt(10) as the exact distance. This is approximately equal to 3.162
If you construct a rectangle with base 3 and height sqrt(10), then you get the blue figure EFGH shown below
The red parallelogram ABCD is what is already given.
The blue and red figures both have the same corresponding side lengths. More technically,
AB = EF = sqrt(10) = 3.162
BC = FG = 3
CD = GH = sqrt(10) = 3.162
HE = DA = 3
The only difference is that the red figure is not a rectangle while the blue figure is a rectangle.
The figures are not congruent as they do not have the same angle measures.
Congruent figures must have all corresponding angles (ie angles that pair up) to be congruent.
Put more simply: the figures must be the same shape and size for us to consider them congruent.
Here's another way to look at it: there is no way we can apply either
- a translation (aka shifting or sliding)
- a rotation, or
- a reflection
More evidence is that the figures have different areas. ABCD has a height of 3 and a base of 3, so its area is base*height = 3*3 = 9. Figure EFGH has base 3 and height sqrt(10) so its area is 3*sqrt(10) = 9.487 approximately, which is slightly larger. In order for the figures to be congruent, they must have the same area.
Congruent figures are effectively the same as saying "the two figures are identical twins". Figures ABCD and EFGH have similar properties in that they have paired sides the same length; however, this isnt enough to fully prove congruence as this example shows.
Once again: to prove quadrilaterals congruent, you need to show that all four sides pair up and have the same length as their other counter part, and also the angles pair up and are congruent to one another.