document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210470>Question 1210470</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the perimeter of triangle XYZ.\r<BR>\n" );
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document.write( "YZ = 10, YP = 6, XQ = 4, XP = 8, PQ = 7 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852802 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=852802\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This problem is solvable by using the property of **tangents drawn from an external point to a circle**.\r<BR>\n" );
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document.write( "The perimeter of $\triangle \text{XYZ}$ is **$28$**.\r<BR>\n" );
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document.write( "## 📐 Tangent Segment Theorem\r<BR>\n" );
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document.write( "The key geometric principle here is the **Tangent Segment Theorem**: If two tangent segments are drawn to a circle from the same external point, then the segments are equal in length.\r<BR>\n" );
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document.write( "In the provided figure, the inscribed circle (incircle) is tangent to the sides of $\triangle \text{XYZ}$ at points $\text{P}$, $\text{Q}$, and a third point, let's call it $\text{R}$, on side $\text{XZ}$.\r<BR>\n" );
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document.write( "* From external point $\mathbf{X}$: The tangent segments are $\mathbf{XQ}$ and $\mathbf{XR}$ (assuming $\text{R}$ is the tangent point on $\text{XZ}$).<BR>\n" );
document.write( "* From external point $\mathbf{Y}$: The tangent segments are $\mathbf{YP}$ and $\mathbf{YQ}$.<BR>\n" );
document.write( "* From external point $\mathbf{Z}$: The tangent segments are $\mathbf{ZP}$ and $\mathbf{ZR}$.\r<BR>\n" );
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document.write( "## 📏 Calculating Side Lengths\r<BR>\n" );
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document.write( "Using the Tangent Segment Theorem and the given lengths:\r<BR>\n" );
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document.write( "1.  **Find the length of the side $\text{YZ}$**:<BR>\n" );
document.write( "    * $\text{YZ}$ is given as $10$.<BR>\n" );
document.write( "    * $\text{YZ} = \text{YP} + \text{PZ}$.<BR>\n" );
document.write( "    * Given $\text{YP} = 6$.<BR>\n" );
document.write( "    * Therefore, $\text{PZ} = \text{YZ} - \text{YP} = 10 - 6 = 4$.\r<BR>\n" );
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document.write( "2.  **Find the lengths of $\text{XQ}$ and $\text{YQ}$**:<BR>\n" );
document.write( "    * Since $\text{YP}$ and $\text{YQ}$ are tangents from $\text{Y}$, we have $\mathbf{\text{YQ} = \text{YP}}$.<BR>\n" );
document.write( "        $$\text{YQ} = 6$$<BR>\n" );
document.write( "    * Since $\text{XQ}$ and $\text{XR}$ (let's use $\text{XR}$ for the tangent to $\text{XZ}$) are tangents from $\text{X}$, we have $\mathbf{\text{XR} = \text{XQ}}$.<BR>\n" );
document.write( "        $$\text{XR} = 4$$\r<BR>\n" );
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document.write( "3.  **Find the length of $\text{ZP}$ and $\text{ZR}$**:<BR>\n" );
document.write( "    * Since $\text{ZP}$ and $\text{ZR}$ are tangents from $\text{Z}$, we have $\mathbf{\text{ZR} = \text{ZP}}$.<BR>\n" );
document.write( "    * We calculated $\text{ZP} = 4$ in step 1.<BR>\n" );
document.write( "        $$\text{ZR} = 4$$\r<BR>\n" );
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document.write( "**Note:** The length $\text{XP} = 8$ and $\text{PQ} = 7$ are extraneous/conflicting if the tangency points are $\text{P}$, $\text{Q}$, and $\text{R}$ on the sides of $\triangle \text{XYZ}$. $\text{X}$, $\text{P}$, and $\text{Q}$ would not be vertices of a known figure, and $\text{XP}$ and $\text{XQ}$ are two tangent lengths from $\text{X}$, which must be equal.\r<BR>\n" );
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document.write( "The only way the problem is solvable is if the tangency points are $\text{Q}$ on $\text{XY}$ and $\text{P}$ on $\text{YZ}$, and the third point is $\text{R}$ on $\text{XZ}$. Given that $\mathbf{\text{XQ} = 4}$ and $\mathbf{\text{XP} = 8}$, the points **$\text{P}$ and $\text{Q}$ cannot both be tangency points** to the same circle from the vertices $\text{X}$ and $\text{Y}$.\r<BR>\n" );
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document.write( "Let's assume the labels in the diagram mean:<BR>\n" );
document.write( "* $\mathbf{XQ}$ and $\mathbf{XR}$ (on $\text{XZ}$) are tangents from $\text{X}$. $\implies \text{XQ} = \text{XR} = 4$.<BR>\n" );
document.write( "* $\mathbf{YP}$ and $\mathbf{YQ}$ are tangents from $\text{Y}$. $\implies \text{YP} = \text{YQ} = 6$.<BR>\n" );
document.write( "* $\mathbf{ZP}$ and $\mathbf{ZR}$ are tangents from $\text{Z}$. $\implies \text{ZP} = \text{ZR}$.\r<BR>\n" );
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document.write( "The length $\mathbf{\text{XP} = 8}$ and $\mathbf{\text{PQ} = 7}$ must be disregarded as they directly contradict the Tangent Segment Theorem for the perimeter calculation, which only requires the tangent lengths from the vertices.\r<BR>\n" );
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document.write( "## ➕ Perimeter Calculation\r<BR>\n" );
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document.write( "The **Perimeter** $(\mathbf{Per})$ of $\triangle \text{XYZ}$ is the sum of its three side lengths:<BR>\n" );
document.write( "$$\text{Per} = \text{XY} + \text{YZ} + \text{XZ}$$\r<BR>\n" );
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document.write( "1.  **Side $\text{XY}$**:<BR>\n" );
document.write( "    $$\text{XY} = \text{XQ} + \text{QY} = 4 + 6 = 10$$<BR>\n" );
document.write( "2.  **Side $\text{YZ}$**:<BR>\n" );
document.write( "    $$\text{YZ} = 10 \text{ (Given)}$$<BR>\n" );
document.write( "3.  **Side $\text{XZ}$**:<BR>\n" );
document.write( "    $$\text{XZ} = \text{XR} + \text{RZ}$$<BR>\n" );
document.write( "    * $\text{XR} = \text{XQ} = 4$<BR>\n" );
document.write( "    * $\text{RZ} = \text{ZP}$<BR>\n" );
document.write( "    * $\text{ZP} = \text{YZ} - \text{YP} = 10 - 6 = 4$<BR>\n" );
document.write( "    * So, $\text{XZ} = 4 + 4 = 8$\r<BR>\n" );
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document.write( "**Total Perimeter:**<BR>\n" );
document.write( "$$\text{Per} = \text{XY} + \text{YZ} + \text{XZ} = 10 + 10 + 8 = 28$$ </SPAN>\n" );
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