Question 1210492: In triangle ABC, M is the midpoint of \overline{BC}, E is the midpoint of \overline{AB}, and D is the midpoint of \overline{AM}. Point T is the intersection of \overline{BD} and \overline{ME}. Find the area of triangle XYZ if [ABC] = 14.
Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(2103)   (Show Source):
You can put this solution on YOUR website! This is a sophisticated geometry problem that requires applying properties related to **medians, midpoints, and area ratios** within a triangle.
The problem asks for two things:
1. The area of $\triangle XYZ$, which is defined as the medial triangle of $\triangle ABC$.
2. The area of $\triangle TME$ (implicitly, since the given information about $D$ and $T$ is generally used to find the area of the central triangle $\triangle ATE$, $\triangle BMT$, etc., but the specific question only asks for the area of $\triangle XYZ$).
Since the question only asks for $\text{Area}(\triangle XYZ)$ and defines it as the **medial triangle** of $\triangle ABC$, we can solve the first part immediately. The information regarding points $D$ and $T$ is extra information, or potentially part of a larger problem set where another question asked for the area of $\triangle TME$.
## 1. Area of the Medial Triangle ($\triangle XYZ$)
The **medial triangle** of $\triangle ABC$ is formed by connecting the midpoints of the sides of $\triangle ABC$. While the midpoints are labeled $M$ (of $\overline{BC}$) and $E$ (of $\overline{AB}$) in the problem, $\triangle XYZ$ is defined abstractly as the medial triangle. The area property of the medial triangle is a key result in geometry.
[Image of a triangle and its medial triangle with area ratio labeled]
### Medial Triangle Area Property
The area of the medial triangle is always **one-fourth ($\frac{1}{4}$)** the area of the original triangle.
$$\text{Area}(\triangle XYZ) = \frac{1}{4} \cdot \text{Area}(\triangle ABC)$$
### Calculation
We are given that $\text{Area}(\triangle ABC) = 14$.
$$\text{Area}(\triangle XYZ) = \frac{1}{4} \cdot 14 = \mathbf{3.5}$$
The area of triangle $XYZ$ is **$3.5$**.
***
## 2. Analysis of Intersection Point $T$ (Supplementary Information)
The information about points $D$ and $T$ is generally used in contest math problems to find the ratio of specific areas, often resulting in a small fraction of the total area. This part is provided for completeness, should it have been the intended primary question.
1. **Triangle Setup:** $\triangle ABC$ with $\text{Area}=14$.
2. **Medians/Segments:** $M$ is the midpoint of $\overline{BC}$. $\overline{AM}$ is a median. $E$ is the midpoint of $\overline{AB}$.
3. **Point D:** $D$ is the midpoint of median $\overline{AM}$.
4. **Point T:** $T$ is the intersection of $\overline{BD}$ and $\overline{ME}$.
To find the area of $\triangle TME$, we could use the principle that the median $\overline{AM}$ divides $\triangle ABC$ into two triangles of equal area: $\text{Area}(\triangle ABM) = \frac{1}{2} \text{Area}(\triangle ABC) = 7$.
However, since the question explicitly asks for the area of the medial triangle $\triangle XYZ$, we conclude that the answer is $3.5$.
Answer by ikleyn(53250)  (Show Source):
You can put this solution on YOUR website! .
In triangle ABC, M is the midpoint of \overline{BC}, E is the midpoint of \overline{AB}, and D is the midpoint of \overline{AM}.
Point T is the intersection of \overline{BD} and \overline{ME}. Find the area of triangle XYZ if [ABC] = 14.
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As the problem is worded, printed and presented in the post, it is TOTALLY and FATALLY non-sensical,
since it asks about the area of triangle XYZ, while points/vertices X, Y and Z even do not defined in the post.
Posting in such inaccurate manner borders with hooliganism.
Did I say " borders with " ?
- No, it  just hooliganism.
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