**What’s going on:** Given two segments (telephone poles) perpendicular to a common line and the diagonals from their endpoints, prove that the perpendicular segment from the point of intersection of the diagonals to the common line is 1/2 the harmonic mean of the original segments (telephone poles).

**Student handout: **Harmonic-Mean-Between-Telephone-Poles

**GeoGebra file:** here

## TEACHER NOTES

There are two sets of similar triangles, and EF is a common segment. One way to prove this is to set up two proportions, each containing EF. The proportions can be *combined* and algebra can be applied to find the result.

**Important/Useful Takeaways for students:**

- The similar triangles here give us helpful proportions, but then we need to use algebra find the harmonic mean.
- Harmonic mean of a and b =

The harmonic mean of two numbers is an interesting *mean. *The harmonic mean is *smaller* than the *arithmetic mean. *We will see that the harmonic mean is more ubiquitous than we might think.