One Way to Estimate Lengths of Diagonal Segments
In the picture below, the length of segment AB is 3 units and the length of the segment BC is 2 units. What about the length of diagonal segment AC? You can determine an estimate for the length of AC by holding a piece of string from A to C and, using A as a center, sweep out a circular arc until it intersects the line running through A and B. (This can also be done with a compass.) We have done that in the picture on the right. In that picture, D is the intersection point. From this, you can see that the length of the diagonal segment AC is between 3 and 4 units, and apparently slightly closer to 4 units.

As you work on dot paper to locate diagonal line segments of specified lengths, there are two rules you must always follow:
 Each line segment you locate must begin and end on a dot.
 You cannot measure using a ruler or other measuring device.
 Sketch a diagonal line segment on the dot paper shown below with
 Length between 2 and 3 units.
 Length between 4 and 5 units.
 Length between 5 and 6 units.
 Describe how you constructed these lines segments.
 The picture below shows a quadrilateral EFGH, embedded in a square.
 You say to your friend, Doubting Thomas: "Quadrilateral EFGH is also a square. It has 4 right angles and all of its sides are equal." Thomas says, "I doubt it." Explain how you would convince Thomas that EFGH has the properties of a square.
 Calculate the exact area of square EFGH.
Explain how you calculated that area.
 Use the area of square EFGH to calculate the exact length of segment EF.
Explain why your method works.
 You recognize that there are other line segments on your dot paper with the same exact length as segment EF.
 You tell Doubting Thomas: "Segments AC and EF have the same exact length. Thomas says, "I doubt it." Explain how you would convince Thomas that you are correct (without using a ruler or compass to measure and without cutting one segment out to place it on top of the other).
 Now sketch a line segment on the dot paper shown below that has length equal to the square root of 26 units. Explain how you calculated this length for the segment.