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I probably should start a new word game called find all the typos! 'Simpel' and rightt' are pretty impressive variations. I guess if you can't spell write rite, then you're not alright! Well, I'm trying to correct these but once they go to a reader, the mistakes are out there for all to see! Haste makes --------------.

Just another one of those square and right triangle problems. Yes, I do have a passion for these types of geometry questions (I've published similar ones previously) and you probably will recognize this one from your own experience. The difference of course is that an investigation takes students from particular cases to a general formula which is related to the harmonic mean of the legs of the right triangle.

Part I

Find the length of the side of the square in Fig. I.

Comments:

This one is eminently guessable and doesn't require the use of similar triangles. Encourage students to justify their conjecture that D and F are midpoints.

Part II

Do the same for Fig. II.

Comments: Ok, the answer is 12/7. You may find students trying a Pythagorean approach, guessing midpoints or assuming special angles. Most students do not look for similar triangle solutions unless they have considerable experience with this or the problem is assigned in that unit!

Part III

Of course, we will now ask students to generalize the result:

If the legs of the right triangle have lengths a and b, show that the side of the indicated square (inscribed square, largest inscribed square with sides parallel to the legs, however you want to describe it!) has length ab/(a+b).