Question 1210473: If ST = 2, QS = 4, and PT = 5, find PQ.
Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(2103)   (Show Source):
You can put this solution on YOUR website! This problem is a classic application of the **Tangent-Secant Theorem** (a form of the Power of a Point Theorem) in circle geometry. Since no figure is provided, we must assume the geometric setup that allows for a unique solution using the given segment lengths.
### 1. Assuming the Geometric Setup
We assume the following configuration:
* $\overline{PT}$ is a **tangent** to a circle at point $T$.
* $\overline{PSQ}$ is a **secant** line intersecting the same circle at points $S$ and $Q$.
* $P$ is the external point from which the tangent and secant originate.
* The length $ST=2$ is **extraneous information** (a common feature in test questions to check if the student uses the correct theorem).
### 2. Applying the Tangent-Secant Theorem
The theorem states that the square of the length of the tangent segment is equal to the product of the length of the external secant segment and the entire secant segment.
$$\mathbf{PT^2 = PS \cdot PQ}$$
From the given information:
* $PT = 5$
* $QS = 4$
* $PQ$ is the entire secant segment: $PQ = PS + QS = PS + 4$
### 3. Calculation
Let $PS = x$. The equation becomes:
$$5^2 = x \cdot (x + 4)$$
$$25 = x^2 + 4x$$
Rearrange into a quadratic equation:
$$x^2 + 4x - 25 = 0$$
Use the quadratic formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, to solve for $x$:
$$x = \frac{-4 \pm \sqrt{4^2 - 4(1)(-25)}}{2(1)}$$
$$x = \frac{-4 \pm \sqrt{16 + 100}}{2}$$
$$x = \frac{-4 \pm \sqrt{116}}{2}$$
Simplify $\sqrt{116} = \sqrt{4 \cdot 29} = 2\sqrt{29}$:
$$x = \frac{-4 \pm 2\sqrt{29}}{2}$$
$$x = -2 \pm \sqrt{29}$$
Since $x$ represents the length $PS$, it must be positive.
$$PS = x = -2 + \sqrt{29}$$
### 4. Finding PQ
The length $PQ$ is the sum of the external segment $PS$ and the internal segment $QS$:
$$PQ = PS + QS$$
$$PQ = (-2 + \sqrt{29}) + 4$$
$$PQ = \mathbf{2 + \sqrt{29}}$$
Answer by ikleyn(53250)  (Show Source):
You can put this solution on YOUR website! .
.
If ST = 2, QS = 4, and PT = 5, find PQ.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Another version of the Artificial Intelligence (the Google Overview) treats the problem differently,
than @CPhill treats it, using different interpretation, and obtains very different answer PQ = 15.
See the link
https://www.google.com/search?q=If+ST+%3D+2%2C+QS+%3D+4%2C+and+PT+%3D+5%2C+find+PQ.&rlz=1C1CHBF_enUS1071US1071&oq=If+ST+%3D+2%2C+QS+%3D+4%2C+and+PT+%3D+5%2C+find+PQ.&gs_lcrp=EgZjaHJvbWUyBggAEEUYOTIHCAEQIRigATIHCAIQIRigATIHCAMQIRigATIHCAQQIRiPAjIHCAUQIRiPAjIHCAYQIRiPAtIBCTk1MTVqMGoxNagCCLACAfEFoBqKcW-un9c&sourceid=chrome&ie=UTF-8
of today, December 2, 2025.
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In Math, there is a golden and immutable rule: the problem formulation must be complete, clear, precise,
and unambiguous. In this case, the points and line segments are introduced, but it is unclear what figure
or configuration they refer to.
In school mathematics, formally speaking, such problems are not accepted and are considered as defective.
Therefore, when I see a stream of such defective problems at this forum last days, which, moreover, are
intentionally made defective, I want to ask: dear sirs, did you fall out of a tree and hit your heads?
And do you need urgent medical attention?
Dear sirs, do you intend to follow these golden and immutable rules of mathematics?
Or have you come to break them?
You must understand that if you undermine these immutable rules, it will sow corruption in the minds
of schoolchildren, and they will not take mathematics seriously at all.
God forbid if your exercises damage the already shaky system of mathematics education in this country.
Then generations to come will curse you for your so called artificial intelligence, and quite rightly so.
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