Triangle Median Problem: Finding AD=1/2AB and BE=1/2EC Relationship

Q: How do I know which point is actually a midpoint?

Look for ratios that equal 1/2! When $ AD = \frac{1}{2}AB $, point D divides AB exactly in half. But $ BE = \frac{1}{2}EC $ means E is not at the midpoint of BC.

Q: Why isn't AE a median if A is a vertex?

Even though AE starts from vertex A, point E is not the midpoint of side BC. Since $ BE = \frac{1}{2}EC $, E divides BC in a 1:2 ratio, not equally.

Q: Can a triangle have more than one median?

Yes! Every triangle has exactly three medians - one from each vertex to the midpoint of the opposite side. They all intersect at the centroid.

Q: What's the difference between a median and just any line segment?

A median has two specific requirements: it must start at a vertex and end at the midpoint of the opposite side. Random line segments don't follow this rule.

Q: How can I double-check my median identification?

Use the two-step check: (1) Does it connect a vertex to the opposite side? (2) Is the endpoint the exact midpoint of that side? Both must be true!

Triangle Medians with Point Location Analysis

Look at the triangle ABC below.

$AD=\frac{1}{2}AB$

$BE=\frac{1}{2}EC$

What is the median in the triangle?

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Step-by-step written solution

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Understand the problem

Look at the triangle ABC below.

$AD=\frac{1}{2}AB$

$BE=\frac{1}{2}EC$

What is the median in the triangle?

Step-by-step solution

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 $DC$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Definition: A median connects a vertex to the midpoint of the opposite side
Identification: Find which point is the midpoint: $AD = \frac{1}{2}AB$ means D is midpoint of AB
Verification: Check that the segment connects vertex C to midpoint D of opposite side AB ✓

Common Mistakes

Avoid these frequent errors

Confusing which point is the midpoint
Don't assume E is a midpoint just because it's marked on the diagram = wrong median identification! The condition $BE = \frac{1}{2}EC$ means E divides BC in ratio 1:2, not equally. Always check the given ratios to identify true midpoints.

Practice Quiz

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Can a triangle have a right angle?

FAQ

Everything you need to know about this question

How do I know which point is actually a midpoint?

Look for ratios that equal 1/2! When $AD = \frac{1}{2}AB$ , point D divides AB exactly in half. But $BE = \frac{1}{2}EC$ means E is not at the midpoint of BC.

Why isn't AE a median if A is a vertex?

Even though AE starts from vertex A, point E is not the midpoint of side BC. Since $BE = \frac{1}{2}EC$ , E divides BC in a 1:2 ratio, not equally.

Can a triangle have more than one median?

Yes! Every triangle has exactly three medians - one from each vertex to the midpoint of the opposite side. They all intersect at the centroid.

What's the difference between a median and just any line segment?

A median has two specific requirements: it must start at a vertex and end at the midpoint of the opposite side. Random line segments don't follow this rule.

How can I double-check my median identification?

Use the two-step check: (1) Does it connect a vertex to the opposite side? (2) Is the endpoint the exact midpoint of that side? Both must be true!

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