Triangle Median Problem: Calculate BC Length Given AD as Median

Q: What exactly is a median in a triangle?

A median is a line segment that connects any vertex of a triangle to the midpoint of the opposite side. Every triangle has exactly 3 medians!

Q: How do I know D is the midpoint of BC?

The problem states that AD is the median. By definition, this means A connects to the midpoint of the opposite side BC, so D must be that midpoint.

Q: Why are BD and DC equal?

Since D is the midpoint of BC, it divides BC into two equal parts. This is what "midpoint" means - the point that splits a line segment exactly in half.

Q: What if BD was a different number?

The method stays the same! If $ BD = 8 $, then $ DC = 8 $ too, so $ BC = 8 + 8 = 16 $. The key is that both segments are always equal.

Q: Can I solve this without knowing about medians?

Not really! Understanding that medians create midpoints is essential. Without this concept, you wouldn't know that D divides BC into two equal parts.

Q: Do all triangles have the same median properties?

Yes! Every triangle - whether scalene, isosceles, or equilateral - has medians that connect vertices to midpoints. This property never changes.

Triangle Medians with Midpoint Properties

AD is the median.

Calculate the length of the side BC.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Calculate side BC

00:03 The entire side equals the sum of its parts

00:09 AD is a median according to the given data, the median bisects the side

00:17 Place BD instead of DC (they are equal because AD is a median)

00:28 Substitute the value of BD according to the given data and solve for BC

00:31 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

AD is the median.

Calculate the length of the side BC.

Step-by-step solution

Given that $AD$ is a median in triangle $\triangle ABC$ and $BD = 6$ , we need to determine the length of side $BC$ .

Since $AD$ is the median, it implies that $D$ is the midpoint of $BC$ . By definition of midpoint, this means:

$BD = DC$ .
Given $BD = 6$ , it follows that $DC = 6$ as well, since $D$ divides $BC$ into two equal parts.

Therefore, the total length of $BC$ can be calculated as:

$BC = BD + DC = 6 + 6 = 12$ .

Thus, the length of side $BC$ is 12.

Final Answer

Key Points to Remember

Essential concepts to master this topic

Definition: A median connects vertex to midpoint of opposite side
Technique: Since $D$ is midpoint, $BD = DC = 6$
Check: Verify midpoint divides side equally: $BC = 6 + 6 = 12$ ✓

Common Mistakes

Avoid these frequent errors

Using median length instead of side segment
Don't confuse the median $AD$ with side segment $BD = 6$ ! Students often try to use the median length as the full side length. Always remember that the median goes from vertex to midpoint, while we need the full side which equals two equal segments.

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

What exactly is a median in a triangle?

A median is a line segment that connects any vertex of a triangle to the midpoint of the opposite side. Every triangle has exactly 3 medians!

How do I know D is the midpoint of BC?

The problem states that AD is the median. By definition, this means A connects to the midpoint of the opposite side BC, so D must be that midpoint.

Why are BD and DC equal?

Since D is the midpoint of BC, it divides BC into two equal parts. This is what "midpoint" means - the point that splits a line segment exactly in half.

What if BD was a different number?

The method stays the same! If $BD = 8$ , then $DC = 8$ too, so $BC = 8 + 8 = 16$ . The key is that both segments are always equal.

Can I solve this without knowing about medians?

Not really! Understanding that medians create midpoints is essential. Without this concept, you wouldn't know that D divides BC into two equal parts.

Do all triangles have the same median properties?

Yes! Every triangle - whether scalene, isosceles, or equilateral - has medians that connect vertices to midpoints. This property never changes.

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