document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210463>Question 1210463</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A straight line \ell divides a triangle \Delta into two congruent triangles.  Select all the statements that must be true.\r<BR>\n" );
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document.write( "(a) If Delta is isoceles, then it is equilateral\r<BR>\n" );
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document.write( "(b) If Delta is right, then it is equilateral\r<BR>\n" );
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document.write( "(c) \ell is parallel to a side of Delta\r<BR>\n" );
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document.write( "(d) \ell is longer than the midpoint of Delta </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852732 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=852732\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This is a fascinating geometry question! For a straight line $\ell$ to divide a triangle $\Delta$ into two **congruent** triangles, $\ell$ must pass through a vertex and the midpoint of the opposite side. Such a line is called a **median** .\r<BR>\n" );
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document.write( "Let $\triangle ABC$ be the original triangle, and let $\ell$ be the line segment $AD$, where $D$ is a point on $BC$. If $\ell$ divides $\triangle ABC$ into two congruent triangles, $\triangle ABD \cong \triangle ACD$, the following must be true:\r<BR>\n" );
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document.write( "1.  **Congruence Proof (SSS or SAS):**<BR>\n" );
document.write( "    * **Side $AD$** is common to both triangles.<BR>\n" );
document.write( "    * If $\triangle ABD \cong \triangle ACD$, then the corresponding sides must be equal. Thus, $AB = AC$ and $BD = CD$.<BR>\n" );
document.write( "2.  **Implications:**<BR>\n" );
document.write( "    * Since $AB = AC$, the original triangle $\triangle ABC$ **must be isosceles**.<BR>\n" );
document.write( "    * Since $BD = CD$, the point $D$ must be the **midpoint** of the side $BC$.<BR>\n" );
document.write( "    * Therefore, the line $\ell$ (segment $AD$) must be the **median** to the base $BC$.\r<BR>\n" );
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document.write( "Now, let's evaluate the given statements based on this required condition ($\triangle ABC$ is isosceles and $\ell$ is the median to the base):\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "## ✅ Statements That Must Be True\r<BR>\n" );
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document.write( "Based on the required condition that $\triangle ABC$ must be isosceles and $\ell$ must be the median to the base, **none** of the given statements are necessarily true. However, let's analyze them for the intended context:\r<BR>\n" );
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document.write( "### (a) If $\Delta$ is isoceles, then it is equilateral\r<BR>\n" );
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document.write( "**False.** The line $\ell$ only requires the two adjacent sides it connects to be equal ($AB = AC$). It does not require the third side ($BC$) to be equal to them. For example, a $9-9-4$ isosceles triangle can be divided into two congruent right triangles by the median to the base, but it is not equilateral.\r<BR>\n" );
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document.write( "### (b) If $\Delta$ is right, then it is equilateral\r<BR>\n" );
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document.write( "**False.** A right triangle (e.g., a $5-12-13$ triangle) is not equilateral. If a right triangle is divided into two congruent triangles, it must be an isosceles right triangle (a $45^\circ-45^\circ-90^\circ$ triangle). Even in that case, it is not equilateral.\r<BR>\n" );
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document.write( "### (c) $\ell$ is parallel to a side of $\Delta$\r<BR>\n" );
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document.write( "**False.** As established, $\ell$ must be a **median** passing through a vertex. A line parallel to a side would create a similar triangle and a trapezoid, which cannot be congruent to each other unless the triangle is degenerate (a line segment).\r<BR>\n" );
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document.write( "### (d) $\ell$ is longer than the midpoint of $\Delta$\r<BR>\n" );
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document.write( "**False.** The term \"**midpoint of $\Delta$**\" is not standard geometric terminology. It might refer to the **centroid** (the intersection of the medians). The length of a median ($\ell$) is just a length, and the centroid is a point, so comparing them is not meaningful.\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "## ❓ Re-evaluating the Problem and Finding the Best Fit\r<BR>\n" );
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document.write( "Given that this is a multiple-choice question where a correct option is typically expected, there might be a typo or misstatement in the options, particularly in (d). Let's consider the properties of the median $\ell$ in this isosceles triangle $\triangle ABC$:\r<BR>\n" );
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document.write( "The line $\ell$ (the median $AD$ to the base $BC$) is also the **altitude** (height) and the **angle bisector** of $\angle BAC$.\r<BR>\n" );
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document.write( "Since the options as written are all strictly false, let's assume option (d) was intended to be a known property of the median. It is possible it was meant to be:<BR>\n" );
document.write( "* \"$\ell$ passes through the **midpoint** of the side it intersects.\" (**True**, as shown above)<BR>\n" );
document.write( "* \"$\ell$ is perpendicular to the base $BC$.\" (**True**, since it's an altitude)\r<BR>\n" );
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document.write( "Since none of the statements as written are true, and the requirement for the split is that $\Delta$ **must be isosceles** and $\ell$ **must be the median to the base**, you must select based on the strict geometric facts.\r<BR>\n" );
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document.write( "**Strictly speaking, none of the options A, B, C, or D are correct.**\r<BR>\n" );
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document.write( "However, if we are forced to choose the statement that is **most related to the properties of $\ell$**:<BR>\n" );
document.write( "* $\ell$ is the median, which involves the **midpoint** of a side.\r<BR>\n" );
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document.write( "Given the typical constraints of the source material, and the high probability of error in the options: **If a diagram suggested $\ell$ was the median, the fact that $\ell$ involves a midpoint is the core geometric feature.**\r<BR>\n" );
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document.write( "Since a definitive choice cannot be made from the given false statements, I must state that **none must be true**.\r<BR>\n" );
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document.write( "* **(a) False.**<BR>\n" );
document.write( "* **(b) False.**<BR>\n" );
document.write( "* **(c) False.**<BR>\n" );
document.write( "* **(d) False.** </SPAN>\n" );
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