Triangle Centroid Practice Problems & Median Intersection

Since line segment DE does not correspond to a full side of any of the triangles present within the given geometry, we conclude that the statement “DE is a side in one of the triangles” is Not true.

To solve the problem of identifying to which side of triangle $ABC$ the median and the altitude are drawn, let's analyze the diagram given for triangle $ABC$.

• We acknowledge that a median is a line segment drawn from a vertex to the midpoint of the opposite side. An altitude is a line segment drawn from a vertex perpendicular to the opposite side.
• Upon reviewing the diagram of triangle $ABC$, line segment $AD$ is a reference term. It appears to meet point $C$ in the middle, suggesting it's a median, but it also forms right angles suggesting it is an altitude.
• Given the placement and orientation of $AD$, it is perpendicular to line $BC$ (the opposite base for the median from $A$). Therefore, this line is both the median and the altitude to side $BC$.

Thus, the side to which both the median and the altitude are drawn is BC.

Therefore, the correct answer to the problem is the side $BC$, corresponding with choice $\text{Option 2: BC}$.

A median of a triangle is a line segment that connects a vertex to the midpoint of the opposite side. In triangle $\triangle ABC$, we need to identify such a median from the diagram provided.

Step 1: Observe the diagram to identify the midpoint of each side.

Step 2: It is given that point $E$ is located on side $AC$. If $E$ is the midpoint of $AC$, then any line from a vertex to point $E$ would be a median.

Step 3: Check line segment $BE$. This line runs from vertex $B$ to point $E$.

Step 4: Since $E$ is labeled as the midpoint of $AC$, line $BE$ is the median of $\triangle ABC$ drawn to side $AC$.

Therefore, the median of the triangle is $BE$ for $AC$.

To solve this problem, we must identify which line segment in triangle ABC is the median.

First, review the definition: a median in a triangle connects a vertex to the midpoint of the opposite side. Now, in triangle ABC:

• Point A represents the vertex.
• Point E lies on line segment AB.
• Line segment EC needs to be checked to see if it connects vertex E to point C.

From the diagram, it appears that E is indeed the midpoint of side AB. Thus, line segment EC connects vertex C to this midpoint.

This fits the definition of a median, verifying that EC is the median line segment in triangle ABC.

Therefore, the solution to the problem is: $\text{EC}$.

A median in a triangle is a line segment connecting a vertex to the midpoint of the opposite side. Here, we need to find such a segment in triangle $\triangle ABC$.

Let's analyze the given conditions:

• $AD = \frac{1}{2}AB$: Point $D$ is the midpoint of $AB$.
• $BE = \frac{1}{2}EC$: Point $E$ is the midpoint of $EC$.

Given that $D$ is the midpoint of $AB$, if we consider the line segment $DC$, it starts from vertex $D$ and ends at $C$, passing through the midpoint of $AB$ (which is $D$), fulfilling the condition for a median.

Therefore, the line segment $DC$ is the median from vertex $A$ to side $BC$.

In summary, the correct answer is the segment $DC$.