MHB Explaining Dimensions of Tanagrams Without Midpoint Assumptions

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The discussion centers on determining the dimensions of Tanagram pieces without making midpoint assumptions. The user has identified the dimensions of the two large congruent isosceles triangles based on the overall square's dimensions of 1 unit by 1 unit. They utilized the Pythagorean theorem to find the lengths of the triangle's legs, concluding that both legs are equal. The next step involves analyzing the square's diagonal to establish further dimensions, leading to the conclusion that the square's dimensions can be derived as (√2)/4. This method allows for the completion of the dimensions for all Tanagram shapes without relying on midpoint assumptions.
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I am currently working on an assignment using Tanagrams. I have the information that I have:
• 2 large, and congruent, isosceles right triangles
• 1 medium isosceles right triangle
• 2 small, and congruent, isosceles right triangles
• 1 square
• 1 parallelogram

The pieces can be rearranged with no gaps or overlapping of shapes into a square with dimensions 1 unit by 1 unit (i.e., the entire area of the square is 1 unit^{2})

I have figured out all the dimensions of the pieces but I need to explain how I came up with them. The only thing is you cannot make midpoint assumptions (e.g. B is the midpoint between A and C). I used this assumption in determining the dimensions so I need a new way to explain the dimensions. I have already explained how I got the 2 large congruent isosceles triangles, but I am stuck.
I have attached a picture of the tanagram

This is what I have so far for my explanation:


The way that I determined the dimensions of each of the 7 shapes was by the information within the task. The information given says that the dimension of the whole square is 1 unit by 1 unit. So for the two large congruent isosceles triangles (segment AJ and JK) the one side is 1 unit because the one side takes up the whole side of the square as you can see by the 1 in red. Next, I tried to determine the dimension of the other two legs. By Theorem 6 of Pythagorean theorem: A triangle is a right triangle if and only if the sum of the squares of the two smaller legs equal the square of the largest leg; so (UCertify, n.a). Next what I did was plugged in 1 for C and squared it which is 1. Since the triangles are isosceles that mean both sides are congruent and equal. So, I divided 1 by 2 and then took the square root to find out what the two sides are which are
 

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What you have done so far is to identify the dimensions of the two big triangles $AFJ$ and $JFK$. I think that you should next look at the square $BDFE$. Draw its diagonal $BF$. You know that $F$ is the centre of the square $ACKJ$, and you know that the angles $BFD$ and $FAJ$ are both $45^\circ$. It follows that $BF$ is parallel to $AJ$, and you should be able to conclude that the length of $BF$ is $\frac12$. Then use Pythagoras to find the lengths of the sides of the square $BDFE$, and deduce that $B$ is indeed the midpoint of $AC.$

[Spelling suggestion: the name of these shapes is Tangrams, with two a's, not three.]
 
So now I can prove the dimensions of square are (√2)/4. With that information I then can fill in the rest of the shapes of the tangrams because I know one length of the square?
 
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