SOLUTION: Hi, I am currently working on creating my own geometry word problems using the Desmos geometry tool, for a school project. My word problem features two congruent, completely equ

Question 1199620: Hi,
I am currently working on creating my own geometry word problems using the Desmos geometry tool, for a school project. My word problem features two congruent, completely equal, oblique triangles which are right next to each other; meaning, the lower right vertex of the left-most triangle connects with the lower left vertex of the right-most triangle. The base of each triangle is 3.71, in the tool's units. But what I noticed is that the distance between the tips of each triangle is the same as the base. That is to say, the distance between the top vertex of one triangle to the other is equal to the length of each triangle's base. Why is this so? I mean, it seems rather intuitive, but I want to know if it has to do with congruency or something. Outside of pure curiosity, I want to know because in my word problem, the student is expected to deduce that the distance between the tips is the same length as the bases, but I can't tell if this is an unreasonable deduction to ask of. If it is, I plan to simply ask the student to make that assumption in my word problem
Thank you so much and I apologize for the lengthy question,
Fabrizio Farfán
Answer by math_tutor2020(3816) (Show Source): You can put this solution on YOUR website!

What you're observing isn't a coincidence.

Consider a right triangle ABC such that segment BC is the base, and point A is directly above B.

Copy triangle ABC and shift it over exactly BC units to the right. This will have points B' and C line up perfectly.

Now connect points A and A' to form a segment. Refer to figure 1 shown below.
This segment is the same length as side BC. We have rectangle ABCA', which is why AA' is congruent to BC.

The last thing to do is nudge point A some amount to the right (see figure 2). The amount doesn't matter. Let's say it's 1 unit to the right. This same nudge is applied to A' as well. Both points move the same amount to the right, which means segment AA' doesn't change in length. The segment is simply translated that amount to the right. Hence AA' is still congruent to BC.

Example Diagrams

and

Another way to look at it: The quadrilateral ABCA' in either diagram is a parallelogram. Meaning we have two sets of parallel opposite sides. One such property is that the opposite sides are congruent. It's similar to a rectangle but the sides can shear and skew.

If you want to prove this using triangle congruences, then focus on triangles ABC and B'A'A. The order of the lettering is important so we can determine how the letters pair up together.

Side note:
GeoGebra was used to make the diagrams shown. It's a tool similar to Desmos.

Yet another way to look at it:
If you shifted triangle ABC exactly BC units to the right, then point A moves the same distance. It must move this distance so it stays attached to the triangle we're moving. Otherwise, it'll transform the second triangle into something that isn't congruent to the first triangle.

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