Depth Of A Dent

Geometry Level 4

I accidentally dropped a large, heavy metal cube onto my nice wood floor. It landed on one of its corners, and made a huge dent. (Okay, so my wood floor isn't "nice." It's balsa. I knew I shouldn't have just gone with the cheapest option...) The triangular hole it made in the surface of the floor has sides of length 68mm, 75mm, and 77mm. The depth of the dent, in mm, can be written as \(\sqrt{n}\), where n is a positive integer. Find n.

Details & Assumptions:

The depth is measured perpendicular to the floor.

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