SOLUTION: A straight line \ell divides a triangle \Delta into two congruent triangles. Select all the statements that must be true. (a) If Delta is isoceles, then it is equilateral

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Question 1210463: A straight line \ell divides a triangle \Delta into two congruent triangles. Select all the statements that must be true.

(a) If Delta is isoceles, then it is equilateral
(b) If Delta is right, then it is equilateral
(c) \ell is parallel to a side of Delta
(d) \ell is longer than the midpoint of Delta
Found 3 solutions by CPhill, ikleyn, mccravyedwin:
Answer by CPhill(2189)   (Show Source): You can put this solution on YOUR website!
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):
---
## ✅ 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.
---
## ❓ 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.**
Answer by ikleyn(53763)   (Show Source): You can put this solution on YOUR website!
.
A straight line \ell divides a triangle \Delta into two congruent triangles. Select all the statements that must be true.

(a) If Delta is isoceles, then it is equilateral
(b) If Delta is right, then it is equilateral
(c) \ell is parallel to a side of Delta
(d) \ell is longer than the midpoint of Delta
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


The statement (d) in the post makes no sense,

because "midpoint" has no length, and the conception of length is inapplicable to "midpoint".

You can not say "\ell is longer than the midpoint of Delta", because these words can not stay together in this connection.


After reading this post, I have very bad impression about ability of its creator
to compose adequate Math problems of adequate level, using adequate Math language.



Answer by mccravyedwin(421)   (Show Source): You can put this solution on YOUR website!
I suspect that this problem was a mistranslation into English, as it could have
been mistranslated from a language where "centroid" could have translated as
"midpoint". Unlike tutor Ikleyn, I never write anything assuming the student did
anything wrong besides mistyping or mistranslating into English.  I think
perhaps the problem should have been stated this way:
 
A straight line segment L divides triangle D into two congruent triangles.
Select all the statements that must be true.

(a) If D is isosceles, then D is equilateral.
(b) If D is right, then D is equilateral.
(c) L is parallel to a side of D.
(d) L is longer than the distance from either endpoint of L to the centroid
    of D.

A straight line segment L divides triangle D into two congruent triangles. 

This means one end of L must be at a vertex of D; for otherwise L would divide D
into a quadrilateral and a triangle, not two triangles. For the two triangles to
be congruent, L must bisect the angle at its vertex, forming two equal right
angles. Also L must be perpendicular to the side opposite that vertex.  Also L
is a common side of the two triangles. Thus D is isosceles and L divides D into
two right triangles.

We check the choices individually to see if they are true:

(a) If D is isosceles, then D is equilateral.

That is not necessarily true, for D's vertex angle could be 90o and the base 
angles could be 45o each. 

(b) If D is right, then D is equilateral.

That could never be true for eq1uilateral triangles have only three 60o
interior angles and no right angles. 

(c) L is parallel to a side of D.

That could not be true for then L would divide D into a triangle and a
quadrilateral, not two triangles. 

(d) L is longer than the distance from either endpoint of L to the centroid
    of D.

This is true because L is the median of D drawn from the apex of isosceles
triangle D. The centroid of a triangle is 2/3 of the distance from a vertex
to the midpoint of the opposite side.  The centroid is a point along L, and not
an endpoint of L, thus its distance from either endpoint is less than the length
of L. 

Answer: (d)

Edwin
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