Triangle Centroid Practice Problems & Median Intersection

In an isosceles triangle, the median to the base is also the height to the base.

That is, side AD forms a 90° angle with side BC.

That is, two right triangles are created.

Therefore, angle ADC is equal to 90 degrees.

To solve the problem of identifying the median of triangle $\triangle ABC$, we follow these steps:

• Step 1: Understand the Definition - A median of a triangle is a line segment that extends from a vertex to the midpoint of the opposite side.
• Step 2: Identify Potential Medians - Examine segments from each vertex to the opposite side. The diagram labels these connections.
• Step 3: Confirm the Median - Specifically check the segment EC in the context of the line segment from vertex $E$ to the side $AC$, and verify it reaches the midpoint of side $AC$.
• Step 4: Verify Against Options - Given choices allow us to consider which point-to-point connection adheres to our criterion for a median. EC is given as one of the choices.

Observation shows: From point $E$ (assumed from the label and position) that line extends directly to point $C$—a crucial diagonal opposite from considered midpoint indications, suggesting it cuts $AC$ evenly, classifying it as a median.

Upon reviewing the given choices, we see that segment $EC$ is listed. Confirming that $EC$ indeed meets at $C$, the midpoint of $AC$, validates that it is a true median.

Therefore, the correct median of $\triangle ABC$ is the segment $EC$.

The problem asks us to confirm if AB is a side of triangle ADB.

Triangle ADB is defined by its vertices, A, D, and B. A triangle is formed when three vertices are connected by three sides.

• Identify vertices: The vertices of the triangle are A, D, and B.
• Identify sides: The triangle's sides should be AB, BD, and DA.
• Observe: From the provided diagram, AB connects vertices A and B.

Therefore, based on the definition of a triangle and observing the connection between components, side AB indeed is a part of triangle ADB.

This confirms that the statement is True.

In geometry, the distance or length of a line segment between two points is the same, regardless of the direction in which it is measured. Consequently, the segments denoted by $BC$ and $CB$ refer to the same segment, both indicating the distance between points B and C.

Hence, the statement "BC = CB" is indeed True.

To solve this problem, since $AD$ is a median of triangle $ABC$, the median divides the opposite side $BC$ into two equal segments.

Given $BD = 4$, this means that $DC$ must also be equal to 4.

Therefore, the length of $DC$ is $4$.