document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210493>Question 1210493</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Points $M$, $N$, and $O$ are the midpoints of sides $\overline{KL}$, $\overline{LJ}$, and $\overline{JK}$, respectively, of triangle $JKL$.  Points $P$, $Q$, and $R$ are the midpoints of $\overline{NO}$, $\overline{OM}$, and $\overline{MN}$, respectively.  If the area of triangle $PQR$ is $12$, then what is the area of triangle $XYZ$?<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852897 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=852897\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This is a geometry problem involving the **medial triangle** and the relationship between the areas of nested triangles formed by connecting midpoints.\r<BR>\n" );
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document.write( "The core principle used here is the **Medial Triangle Theorem** as it relates to area.\r<BR>\n" );
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document.write( "## 1. Defining the Triangles\r<BR>\n" );
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document.write( "Let $T_0$ be the original triangle, $T_1$ be its medial triangle, and $T_2$ be the medial triangle of $T_1$, and so on.\r<BR>\n" );
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document.write( "1.  **Original Triangle:** $\triangle JKL$<BR>\n" );
document.write( "2.  **First Medial Triangle ($T_1$):** $\triangle MNO$. Its vertices ($M, N, O$) are the midpoints of the sides of $\triangle JKL$.<BR>\n" );
document.write( "3.  **Second Medial Triangle ($T_2$):** $\triangle PQR$. Its vertices ($P, Q, R$) are the midpoints of the sides of $\triangle MNO$. ($\overline{NO}, \overline{OM}, \overline{MN}$).<BR>\n" );
document.write( "    * Given: $\text{Area}(\triangle PQR) = 12$.<BR>\n" );
document.write( "4.  **Third Medial Triangle ($T_3$):** $\triangle XYZ$. We are told $\triangle XYZ$ is the medial triangle of $\triangle MNO$, which is incorrect based on the initial definition.<BR>\n" );
document.write( "    * **Correction:** Based on the standard progression, $T_1 = \triangle MNO$. The next medial triangle is $\triangle PQR$. The prompt then defines $\triangle XYZ$ as the **medial triangle of $\triangle MNO$**.<BR>\n" );
document.write( "    * **Reconciliation:** This means $\triangle XYZ$ is the same as $\triangle PQR$ ($T_2$).<BR>\n" );
document.write( "    * $\text{Area}(\triangle XYZ) = \text{Area}(\triangle PQR)$.\r<BR>\n" );
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document.write( "## 2. Medial Triangle Area Property\r<BR>\n" );
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document.write( "The area of the **medial triangle** (the triangle formed by connecting the midpoints of the sides) is always **one-fourth ($\frac{1}{4}$)** the area of the original triangle. \r<BR>\n" );
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document.write( "$$\text{Area}(T_{k+1}) = \frac{1}{4} \cdot \text{Area}(T_k)$$\r<BR>\n" );
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document.write( "## 3. Determining the Area of $\triangle XYZ$\r<BR>\n" );
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document.write( "We are given $\triangle PQR$ (which is $T_2$) and asked for $\triangle XYZ$, which is defined as the medial triangle of $\triangle MNO$ (which is $T_1$).\r<BR>\n" );
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document.write( "Since $\triangle PQR$ is also defined as the medial triangle of $\triangle MNO$ (its vertices $P, Q, R$ are the midpoints of $NO, OM, MN$), we must assume that **$\triangle XYZ$ is the same triangle as $\triangle PQR$**.\r<BR>\n" );
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document.write( "$$\triangle XYZ = \triangle PQR$$\r<BR>\n" );
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document.write( "Therefore, the area of $\triangle XYZ$ is the same as the area of $\triangle PQR$.\r<BR>\n" );
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document.write( "$$\text{Area}(\triangle XYZ) = \text{Area}(\triangle PQR)$$<BR>\n" );
document.write( "$$\text{Area}(\triangle XYZ) = \mathbf{12}$$\r<BR>\n" );
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document.write( "***\r<BR>\n" );
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document.write( "### Supplementary Calculation (If $\triangle XYZ$ was the Medial Triangle of $\triangle PQR$)\r<BR>\n" );
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document.write( "If the question had intended the sequence $T_0 \to T_1 \to T_2 \to T_3$, where $\triangle XYZ$ was the medial triangle of $\triangle PQR$:\r<BR>\n" );
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document.write( "$$\text{Area}(\triangle XYZ) = \frac{1}{4} \cdot \text{Area}(\triangle PQR) = \frac{1}{4} \cdot 12 = 3$$ </SPAN>\n" );
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