Triangle Median Problem: Find the Ratio of BC to DC

Q: What exactly does a median do in a triangle?

A median is a line segment from any vertex to the midpoint of the opposite side. It always divides that side into two equal parts, no matter what type of triangle you have!

Q: Why isn't the ratio BC:DC equal to 1:1?

Because BC is the whole side while DC is only half of it! Since the median creates $ BD = DC $, we have $ BC = BD + DC = DC + DC = 2DC $.

Q: Does this ratio work for any triangle with a median?

Absolutely! This 2:1 ratio between the whole side and half the side is true for every triangle, regardless of its shape or size. It's a fundamental property of medians.

Q: What if the problem asked for DC:BC instead?

Just flip the ratio! If $ BC:DC = 2:1 $, then $ DC:BC = 1:2 $. Always pay attention to which measurement comes first in the question.

Median Properties with Ratio Calculation

In front of you the next triangle:

Given AD median in the triangle.

Find the ratio of the length of the side BC and the length of DC.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Determine the ratio between BC and DC

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

00:18 The segments are equal according to the given data

00:26 Substitute in DC for BD

00:37 Determine the ratio between BC and DC

00:47 Simplify wherever possible

00:51 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

In front of you the next triangle:

Given AD median in the triangle.

Find the ratio of the length of the side BC and the length of DC.

Step-by-step solution

In triangle $ABC$ , $AD$ is a median from $A$ to side $BC$ . By definition, a median divides the opposite side into two equal segments: $BD = DC$ . Therefore, the total length of side $BC$ is the sum of $BD$ and $DC$ , which simplifies to $BC = BD + DC = 2 \times DC$ .

We are tasked with finding the ratio $BC : DC$ . Given that $BC = 2 \times DC$ , we conclude the ratio is $2:1$ .

Therefore, the solution to the problem is $2:1$ .

Final Answer

2:1

Key Points to Remember

Essential concepts to master this topic

Definition: A median divides the opposite side into two equal segments
Technique: Since $BD = DC$ , then $BC = 2 \times DC$
Check: Verify ratio by confirming $BC:DC = 2DC:DC = 2:1$ ✓

Common Mistakes

Avoid these frequent errors

Confusing which segments are equal in a median
Don't think BD:DC equals 1:1 means BC:DC also equals 1:1 = wrong ratio! This ignores that BC contains both BD and DC segments. Always remember BC = BD + DC, so if BD = DC, then BC = 2 × DC.

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 does a median do in a triangle?

A median is a line segment from any vertex to the midpoint of the opposite side. It always divides that side into two equal parts, no matter what type of triangle you have!

Why isn't the ratio BC:DC equal to 1:1?

Because BC is the whole side while DC is only half of it! Since the median creates $BD = DC$ , we have $BC = BD + DC = DC + DC = 2DC$ .

Does this ratio work for any triangle with a median?

Absolutely! This 2:1 ratio between the whole side and half the side is true for every triangle, regardless of its shape or size. It's a fundamental property of medians.

How can I remember this concept?

Think of it like a pizza slice! If you cut a slice exactly in half, the whole slice is twice as big as each half slice. Same idea: whole side = 2 × half side.

What if the problem asked for DC:BC instead?

Just flip the ratio! If $BC:DC = 2:1$ , then $DC:BC = 1:2$ . Always pay attention to which measurement comes first in the question.

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