document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210473>Question 1210473</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If ST = 2, QS = 4, and PT = 5, find PQ. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852835 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2138)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=852835\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 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.\r<BR>\n" );
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document.write( "### 1. Assuming the Geometric Setup\r<BR>\n" );
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document.write( "We assume the following configuration:<BR>\n" );
document.write( "* $\overline{PT}$ is a **tangent** to a circle at point $T$.<BR>\n" );
document.write( "* $\overline{PSQ}$ is a **secant** line intersecting the same circle at points $S$ and $Q$.<BR>\n" );
document.write( "* $P$ is the external point from which the tangent and secant originate.<BR>\n" );
document.write( "* The length $ST=2$ is **extraneous information** (a common feature in test questions to check if the student uses the correct theorem).\r<BR>\n" );
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document.write( "### 2. Applying the Tangent-Secant Theorem\r<BR>\n" );
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document.write( "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.\r<BR>\n" );
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document.write( "$$\mathbf{PT^2 = PS \cdot PQ}$$\r<BR>\n" );
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document.write( "From the given information:<BR>\n" );
document.write( "* $PT = 5$<BR>\n" );
document.write( "* $QS = 4$<BR>\n" );
document.write( "* $PQ$ is the entire secant segment: $PQ = PS + QS = PS + 4$\r<BR>\n" );
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document.write( "### 3. Calculation\r<BR>\n" );
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document.write( "Let $PS = x$. The equation becomes:<BR>\n" );
document.write( "$$5^2 = x \cdot (x + 4)$$<BR>\n" );
document.write( "$$25 = x^2 + 4x$$\r<BR>\n" );
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document.write( "Rearrange into a quadratic equation:<BR>\n" );
document.write( "$$x^2 + 4x - 25 = 0$$\r<BR>\n" );
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document.write( "Use the quadratic formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, to solve for $x$:<BR>\n" );
document.write( "$$x = \frac{-4 \pm \sqrt{4^2 - 4(1)(-25)}}{2(1)}$$<BR>\n" );
document.write( "$$x = \frac{-4 \pm \sqrt{16 + 100}}{2}$$<BR>\n" );
document.write( "$$x = \frac{-4 \pm \sqrt{116}}{2}$$\r<BR>\n" );
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document.write( "Simplify $\sqrt{116} = \sqrt{4 \cdot 29} = 2\sqrt{29}$:<BR>\n" );
document.write( "$$x = \frac{-4 \pm 2\sqrt{29}}{2}$$<BR>\n" );
document.write( "$$x = -2 \pm \sqrt{29}$$\r<BR>\n" );
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document.write( "Since $x$ represents the length $PS$, it must be positive.<BR>\n" );
document.write( "$$PS = x = -2 + \sqrt{29}$$\r<BR>\n" );
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document.write( "### 4. Finding PQ\r<BR>\n" );
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document.write( "The length $PQ$ is the sum of the external segment $PS$ and the internal segment $QS$:<BR>\n" );
document.write( "$$PQ = PS + QS$$<BR>\n" );
document.write( "$$PQ = (-2 + \sqrt{29}) + 4$$<BR>\n" );
document.write( "$$PQ = \mathbf{2 + \sqrt{29}}$$ </SPAN>\n" );
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