Question

1. why is the? in the proportion below equal to qr?

\\( \frac { q t } { s q } = \frac { s q } {? } \\)
\\( \overline { s q } \\) is the geometric mean between the hypotenuse and the closest adjacent segment of the hypotenuse.
\\( \overline { s q } \\) is the shortest leg and so is the geometric mean between the entire leg and the hypotenuse \\( \overline { q r } \\).
\\( \overline { q r } \\) is the hypotenuse and thus the hypotenuse rule of geometric means applies.
\\( \overline { q r } \\) is the hypotenuse and is opposite the angle s in the right triangle qsr.

Explanation:

In right triangle QSR with altitude ST to hypotenuse QR, the leg SQ is the geometric mean of the hypotenuse QR and the adjacent segment QT of the hypotenuse. This gives the proportion $\frac{QT}{SQ} = \frac{SQ}{QR}$, so the "?" is QR. The first option correctly states that SQ (a leg) is the geometric mean between the hypotenuse (QR) and its closest adjacent segment (QT).

Answer:

$\overline{SQ}$ is the geometric mean between the hypotenuse and the closest adjacent segment of the hypotenuse.