An octagon is formed by joining the points (7,0), (5,5), (0,7), (-5,5), (-7,0), (-5,-5), (0,-7), (5,-5) and (7,0). The octagon is regular.

I have used proof by exhaustion and got that all sides are square root of 29. Then sketched it, and all sides were also same length. But the book says the conjecture's false...

I believe a regular octagon has all sides same side and all angles are the same, which should fallow from all length being the same.

Help!

sqrt (2^2 + 5^2) = sqrt 29 agreed

the distance from -5,5 to 5,5 is 10

the distance from 0,-7 to 7,0
is 7

Oh my :( not really regular
I guess you will have to make a model with 8 equal sides and see if you can sqwish it :)
I think you can.

IF A (0,-7) and B (7,0)

AB = Sqrt of (0-7)^2 + (-7-0) and that is sqrt of 7^2 + 7^2 not 7?

Okay but a regular octagon is meant to have all sides of the same length. (7,0) to (5,-5) is a side and (5,-5) to (0,-7) is a side. (7,0) to (-7,0) is not a side, so why would it matter? I don't get it...

If you can squish it, it is no longer regular. The sides may be the same but not the angles.

Think about a square with hinges at the corners
if you squish it into a parallelogram, it still has 4 equal sides but is no longer a regular polygon.

It seems like you've done some great work so far in trying to prove that the given octagon is regular. Let's go through the process step-by-step to see where the issue might be.

First, you mentioned that you used proof by exhaustion to calculate the length of all the sides and found it to be the square root of 29. This is an excellent starting point. However, to prove that the octagon is regular, you need to show that all the sides have the same length.

To do this, you should use the distance formula to find the length of each side. The distance formula between two points (x1, y1) and (x2, y2) is √[(x2 - x1)^2 + (y2 - y1)^2].

For example, the length of the line segment joining (7, 0) and (5, 5) is:

√[(5 - 7)^2 + (5 - 0)^2] = √[4 + 25] = √29.

Now repeat this calculation for all other pairs of consecutive points to find the lengths of all the sides. If you find that all the sides have the same length, then you would have proved that the octagon is regular.

However, if you find that at least one side has a different length, then the octagon is not regular, as the book claimed. In that case, there might be an error in your calculations or some discrepancy in the given information.

Double-check your calculations and make sure you haven't misread any coordinates or made any arithmetic mistakes. If you're confident in your calculations but still disagree with the book, it may be worth seeking clarification from your teacher or consulting other resources for further assistance.

Remember, proof by exhaustion is a valid method, but it can be time-consuming and prone to errors. Be patient and meticulous in your calculations, and don't hesitate to seek additional help if needed.