- Thread starter s10007
- Start date

- Joined
- Nov 12, 2017

- Messages
- 11,270

Essentially, though, you will be doing the same work with letters replacing the numbers, so the answer will be an expression.

- Joined
- Jan 27, 2012

- Messages
- 7,453

Set up a coordinate system with x- axis along the center wall and origin in the middle of that center wall.

The walls of the room will be at x= -50, x= 50, y= -40, and y= 40. The four corners of the wall are at (-50, 40), (-50, -40), (50, -40) and (50, 40). The ends of the center wall are at (-35, 0) and (35, 0).

The "path" around the room makes turns at the midpoint of the line from (35, 0) TO (50, 40), the midpoint or the line from (-35, 0) to (-50, 40), the midpoint of the line from (-35, 0) to (-50, -40), and the midpoint of the line from (35, 0) to (50, -40), Those midpoints are (42.5, 20), (-42.5, 20), (-42.5, -20), and (42.5, -20). The distance from (42.5, 20) to (-42.5, 20) is 85 and the distance from (-42.5, 20) to (-42.5, -20) is 40. The other two distances are, of course, the same so the length of the circuit is 2(85+ 40)= 2(125)= 250 ft. just as you say!

- Joined
- Nov 12, 2017

- Messages
- 11,270

Good. Now suppose that the 80 and 100 were switched, and answer the question, "Does your answer depend on which sides the wall is parallel to?"

Then do the exact same work, but with the letters a, b, and x in place of the numbers 80, 100, and 50. The result is interesting, and will confirm your numerical answers.

One question from me, though: why are you using "conjectures"? That means statements that have been

- Joined
- Nov 12, 2017

- Messages
- 11,270

So do this:

I shouldn't need to say this over and over.From the triangle midsegment [theorem], I know the widths of the path will be half ofa, so __ and __. From the trapezoid midsegment [theorem] the length of the path will be the average of the bases of the trapezoid, so (b+x)/2 or __ and __. ...