answer it---**Extraction Content:**
**Question Stem:**
In the diagram below of circle O, chords RT and QS intersect at M. Secant PTR and tangent PS are drawn to circle O. The length of RM is two more than the length of TM, QM = 2, SM = 12, and PT = 8.
**Given Information:**
RM is two more than TM (RM = TM + 2)
QM = 2
SM = 12
PT = 8
**Question:**
The length of PS is _____.
**Diagram Description:**
* **Type:** Geometric diagram of a circle.
* **Main Elements:**
* A circle centered at point O.
* Two chords, RT and QS, intersecting at point M inside the circle.
* A secant line PTR originating from point P outside the circle, passing through points T and R on the circle in that order.
* A tangent line PS originating from point P outside the circle and tangent to the circle at point S.
* Points labeled: O (center of the circle), R, T, Q, S (points on the circle), M (intersection of chords RT and QS), P (external point).
* Segments are labeled implicitly by the points defining them: RM, MT, RT (chord), QM, MS, QS (chord), PT, TR, PR (secant), PS (tangent).
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We have a circle with center O. Two chords RT and QS intersect at point M inside the circle. From external point P, we have secant PTR and tangent PS. We're given that RM is two more than TM, QM equals 2, SM equals 12, and PT equals 8. We need to find the length of PS.
We'll use the intersecting chords theorem, which states that when two chords intersect inside a circle, the products of their segments are equal. So RM times TM equals QM times SM. Given that RM equals TM plus 2, QM equals 2, and SM equals 12, we substitute to get TM plus 2 times TM equals 2 times 12. This gives us TM squared plus 2TM equals 24, or TM squared plus 2TM minus 24 equals zero. Factoring, we get TM plus 6 times TM minus 4 equals zero. Since TM must be positive, TM equals 4, and therefore RM equals 6.
Now we calculate the secant segments. From our previous calculation, we know TM equals 4 and RM equals 6. Therefore, the chord RT equals RM plus TM, which is 6 plus 4, equals 10. We're given that PT equals 8. The total secant length PR equals PT plus TR, which is 8 plus 10, equals 18. So we have the external segment PT equals 8 and the total secant length PR equals 18.
Now we apply the tangent-secant theorem. When a tangent and secant are drawn from the same external point, the square of the tangent length equals the product of the secant's external segment and the total secant length. So PS squared equals PT times PR. We know PT equals 8 and PR equals 18. Substituting these values, PS squared equals 8 times 18, which equals 144. Taking the square root, PS equals 12.
To summarize our solution: We used the intersecting chords theorem to find that TM equals 4 and RM equals 6. We then calculated that chord RT equals 10 and the total secant PR equals 18. Finally, we applied the tangent-secant theorem, where PS squared equals PT times PR, giving us PS squared equals 144, so PS equals 12.