Let A and B be points in the upper half-plane, and let P and Q be the ends of the u-line through A and B. We define the cross-ratio  by

Figure 23

We then define the length under the upper half-plane model  by

We interpret the u-segment AB and CD to be length-congruent under the upper half-plane model if .

Given:

Two points A and B in the upper half-plane.

To Construct:

The distance between A and B under the upper half-plane model.

Constructions:

Use the tool ‘Distance and Length’ in Cabri Geometry II to measure the lengths of , ,  and . Then use the tool ‘Calculator’ to calculate the value of  by the above equations.

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