Why is the ? in the proportion below equal to QR? Explain why. A) SQ is the geometric mean between the hypotenuse and the closest adjacent segment of the hypotenuse.
B) SQ is the shortest leg and so is the geometric mean between the entire leg and the hypotenuse.
C) SQ is the hypotenuse and is opposite the angle S in the right triangle QSR.
D) SQ is the hypotenuse and thus the Hypotenuse Rule of Geometric Means applies.

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  A) SQ is the geometric mean between the hypotenuse and the closest adjacent segment of the hypotenuse.

Step-by-step explanation:

In this geometry, all of the right triangles are similar. That means corresponding sides have the same ratio (are proportional).

Here, SQ is the hypotenuse of ΔSQT and the short side of ΔRQS.

Those two triangles are similar, so we can write ...

  (short side)/(hypotenuse) = QT/SQ = QS/RQ

In the above proportion, we have used the vertices in the same order they appear in the similarity statement (ΔSQT ~ ΔRQS). Of course, the names can have the vertices reversed:

  QT/SQ = SQ/QR . . . . . QS = SQ, RQ = QR

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When this is rewritten to solve for SQ, we get ...

  SQ² = QR·QT

  SQ = √(QR·QT) . . . . SQ (short side) is the geometric mean of the hypotenuse and the short segment.