Triangle Median Problem: Finding the 2X:X Ratio in Triangle ABC

Q: How can I tell if point D is the midpoint of BC?

Measure or calculate the distances: CD and DB must be equal. If they're marked with hash marks or given equal lengths, then D is the midpoint and AD would be a median.

Q: What if the diagram shows measurements like 2X and X?

Those measurements tell you the segments are not equal! Since CD = 2X and DB = X, point D is not the midpoint of BC, so AD cannot be a median.

Q: How many medians does every triangle have?

Every triangle has exactly 3 medians - one from each vertex to the midpoint of the opposite side. They all meet at a point called the centroid.

Triangle Median Identification with Misleading Diagrams

What is the median of triangle ABC.

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Step-by-step written solution

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Understand the problem

What is the median of triangle ABC.

Step-by-step solution

In this problem, we must determine if any of the line segments drawn within triangle ABC represent a median. A median is defined as a line segment extending from a vertex to the midpoint of the opposite side.

Upon examining the geometry of triangle ABC presented in the diagram:

The segment extending from A to the base BC and those depicted from B or C should be checked if they connect to a midpoint on the opposite side.
To be considered a median, a line from a vertex must bisect the opposite side into two equal lengths.

None of the segments drawn directly bisect the opposite sides they connect to, as evidenced by either lack of midpoint marking or unequal line segment sections along BC, CA, or AB.

Therefore, after careful inspection, there is no median shown in the given diagram.

Final Answer

There is no median shown.

Key Points to Remember

Essential concepts to master this topic

Median Definition: Line from vertex to midpoint of opposite side
Technique: Check if point D bisects BC: CD = DB required
Verification: Measure both segments - if unequal, not a median ✓

Common Mistakes

Avoid these frequent errors

Assuming any line from vertex to opposite side is a median
Don't think AD is automatically a median just because it goes from A to side BC = wrong identification! The line must specifically reach the midpoint, not just any point. Always verify that the endpoint divides the opposite side into two equal parts.

Practice Quiz

Test your knowledge with interactive questions

Is the straight line in the figure the height of the triangle?

FAQ

Everything you need to know about this question

How can I tell if point D is the midpoint of BC?

Measure or calculate the distances: CD and DB must be equal. If they're marked with hash marks or given equal lengths, then D is the midpoint and AD would be a median.

What if the diagram shows measurements like 2X and X?

Those measurements tell you the segments are not equal! Since CD = 2X and DB = X, point D is not the midpoint of BC, so AD cannot be a median.

Can a triangle have lines that aren't medians?

Absolutely! Triangles can have many other special lines like altitudes (perpendicular to sides), angle bisectors, or just random segments. Not every line is a median.

How many medians does every triangle have?

Every triangle has exactly 3 medians - one from each vertex to the midpoint of the opposite side. They all meet at a point called the centroid.

What should I do if no median is shown in the diagram?

That's a valid answer! Don't force yourself to pick a line segment if none actually meets the median definition. Carefully check each option against the requirements.

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