document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210464>Question 1210464</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find AE<BR>\n" );
document.write( "BE = 30, PE = 18, CP = 10  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852729 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2189)</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=852729\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The problem can be solved by applying the **Intersecting Chords Theorem** to the circle that passes through the four points $A, B, C, E$.\r<BR>\n" );
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document.write( "## 🎯 Finding AE using the Intersecting Chords Theorem\r<BR>\n" );
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document.write( "The Intersecting Chords Theorem states that if two chords, $AC$ and $BE$, intersect at a point $P$ inside a circle, then the product of the segments of one chord equals the product of the segments of the other chord.\r<BR>\n" );
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document.write( "In this case, we have two chords, $AC$ and $BE$, intersecting at $P$.\r<BR>\n" );
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document.write( "1.  **Identify the Segments and Given Lengths:**<BR>\n" );
document.write( "    * Chord $AC$ is divided into segments $AP$ and $CP$.<BR>\n" );
document.write( "    * Chord $BE$ is divided into segments $BP$ and $PE$.<BR>\n" );
document.write( "    * We are given:<BR>\n" );
document.write( "        * $BE = 30$ (Length of the entire chord)<BR>\n" );
document.write( "        * $PE = 18$<BR>\n" );
document.write( "        * $CP = 10$\r<BR>\n" );
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document.write( "2.  **Calculate the length of the segment BP:**<BR>\n" );
document.write( "    The segment $BP$ is the difference between the entire chord $BE$ and the segment $PE$.<BR>\n" );
document.write( "    $$BP = BE - PE$$<BR>\n" );
document.write( "    $$BP = 30 - 18 = 12$$\r<BR>\n" );
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document.write( "3.  **Apply the Intersecting Chords Theorem:**<BR>\n" );
document.write( "    The theorem states:<BR>\n" );
document.write( "    $$AP \cdot CP = BP \cdot PE$$\r<BR>\n" );
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document.write( "4.  **Solve for the unknown segment AP:**<BR>\n" );
document.write( "    Substitute the known values into the equation:<BR>\n" );
document.write( "    $$AP \cdot 10 = 12 \cdot 18$$<BR>\n" );
document.write( "    $$10 \cdot AP = 216$$<BR>\n" );
document.write( "    $$AP = \frac{216}{10} = 21.6$$\r<BR>\n" );
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document.write( "5.  **Calculate the length of AE:**<BR>\n" );
document.write( "    The question asks for the length of $AE$. However, in a standard diagram where chords $AC$ and $BE$ intersect at $P$, $AE$ is a **non-chord segment** (it's a side of $\triangle APE$).\r<BR>\n" );
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document.write( "    **Assuming the request meant the length of the chord AC, as it uses all given information:**\r<BR>\n" );
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document.write( "    * If the request was for $AC$:<BR>\n" );
document.write( "        $$AC = AP + CP$$<BR>\n" );
document.write( "        $$AC = 21.6 + 10 = \mathbf{31.6}$$\r<BR>\n" );
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document.write( "    **Assuming the points form a cyclic quadrilateral $ABCE$ and the question truly asks for $AE$:**\r<BR>\n" );
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document.write( "    We would need more information (like an angle or the radius of the circle) to find the length of $AE$ using the Law of Cosines on $\triangle APE$. The given lengths $AP=21.6, PE=18$ are sides, but the angle $\angle APE$ is missing.\r<BR>\n" );
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document.write( "---<BR>\n" );
document.write( "## ⭐ Finding AE using Power of a Point Theorem (Alternative)\r<BR>\n" );
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document.write( "If the points $A, B, C, E$ form a cyclic quadrilateral and the lines $AC$ and $BE$ are extended to intersect **outside** the circle at a point $P$ (the **Secant-Secant Theorem**), the relationship is different:\r<BR>\n" );
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document.write( "$$PA \cdot PC = PE \cdot PB$$\r<BR>\n" );
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document.write( "Given the phrasing and the internal intersection formula, it is overwhelmingly likely that **$AC$ was the intended segment to find.**\r<BR>\n" );
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document.write( "**Based on the only calculation that utilizes all given numbers through a standard circle theorem, the answer for the chord length AC is 31.6.** If the diagram implies a different theorem (like Secant-Tangent or Secant-Secant), please provide the configuration.\r<BR>\n" );
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document.write( "**Final Answer for the calculated segment AP:**<BR>\n" );
document.write( "$$AP = 21.6$$\r<BR>\n" );
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document.write( "**Final Answer for the full chord AC (likely intended):**<BR>\n" );
document.write( "$$AC = 31.6$$\r<BR>\n" );
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document.write( "Without a diagram, and given the ambiguous request for $AE$ vs. the solvable segments $AP$ and $AC$, the most complete answer is to provide the length of $AP$ and the probable intended length of $AC$.\r<BR>\n" );
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document.write( "Since the question asks for $AE$ and $AE$ cannot be found with the given information, I will provide the two relevant segments calculated: $AP$ and the full chord $AC$.\r<BR>\n" );
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document.write( "**Length of segment $AP$ is $\mathbf{21.6}$.**<BR>\n" );
document.write( "**Length of chord $AC$ is $\mathbf{31.6}$.** </SPAN>\n" );
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