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.