Question 1210463
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** .

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:

1.  **Congruence Proof (SSS or SAS):**
    * **Side $AD$** is common to both triangles.
    * If $\triangle ABD \cong \triangle ACD$, then the corresponding sides must be equal. Thus, $AB = AC$ and $BD = CD$.
2.  **Implications:**
    * Since $AB = AC$, the original triangle $\triangle ABC$ **must be isosceles**.
    * Since $BD = CD$, the point $D$ must be the **midpoint** of the side $BC$.
    * Therefore, the line $\ell$ (segment $AD$) must be the **median** to the base $BC$.

Now, let's evaluate the given statements based on this required condition ($\triangle ABC$ is isosceles and $\ell$ is the median to the base):

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## ✅ Statements That Must Be True

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:

### (a) If $\Delta$ is isoceles, then it is equilateral

**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.

### (b) If $\Delta$ is right, then it is equilateral

**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.

### (c) $\ell$ is parallel to a side of $\Delta$

**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).

### (d) $\ell$ is longer than the midpoint of $\Delta$

**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.

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## ❓ Re-evaluating the Problem and Finding the Best Fit

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 $\triangle ABC$:

The line $\ell$ (the median $AD$ to the base $BC$) is also the **altitude** (height) and the **angle bisector** of $\angle BAC$.

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:
* "$\ell$ passes through the **midpoint** of the side it intersects." (**True**, as shown above)
* "$\ell$ is perpendicular to the base $BC$." (**True**, since it's an altitude)

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.

**Strictly speaking, none of the options A, B, C, or D are correct.**

However, if we are forced to choose the statement that is **most related to the properties of $\ell$**:
* $\ell$ is the median, which involves the **midpoint** of a side.

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.**

Since a definitive choice cannot be made from the given false statements, I must state that **none must be true**.

* **(a) False.**
* **(b) False.**
* **(c) False.**
* **(d) False.**