Compress, Decompress, Measure the Damage

A service stores each user as a two-number record $(x, y)$ and wants to compress every record down to a single number using PCA. The pipeline is fitted once, on these five training records:

Fitting means: centre the training records, build their $2 \times 2$ covariance matrix, and take the unit eigenvector of its largest eigenvalue as the PC1 direction.

A new user record then arrives at $(12,\,9)$. It is not used to fit anything — the mean and the covariance matrix stay exactly as the five training records defined them. The new record is pushed through the finished pipeline:

1. Centre — subtract the training mean.
2. Compress — project the centred record onto the unit PC1 direction, keeping the single resulting number.
3. Decompress — multiply that number back by the same unit PC1 direction, then add the training mean back on.

How far is the decompressed point from the original record $(12, 9)$?

Give the Euclidean distance, rounded to two decimal places. Carry at least four decimal places through the intermediate steps and round only at the end.