Triangle Area with Median: Finding Area of ADC when ABC = 32

Q: What exactly is a median in a triangle?

A median is a line segment that connects a vertex to the midpoint of the opposite side. In this problem, AD connects vertex A to point D, which is the midpoint of side BC.

Q: Why do medians always create equal areas?

Because the median divides the base into two equal parts! Since both resulting triangles share the same height from the apex, and their bases are equal, their areas must be equal too.

Q: How is this different from an altitude?

An altitude is perpendicular to the base and creates right angles. A median goes to the midpoint but isn't necessarily perpendicular. They're completely different concepts!

Q: What if the triangle isn't drawn to scale?

It doesn't matter! The median property works for any triangle - scalene, isosceles, or equilateral. The visual appearance doesn't change the mathematical relationship.

Q: Can I use this property for other medians too?

Absolutely! Every median in any triangle divides it into two triangles of equal area. This is a fundamental property that always works.

Q: How do I remember this isn't half the perimeter?

Focus on what medians do: they divide areas, not lengths. The area gets split in half because the base gets split in half, while the height stays the same.

Triangle Medians with Area Division

In front of you the next triangle:

Since AD is the median

Since the area of the triangle ABC is equal to 32.

Find the area of the triangle ADC.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:09 Let's find the area of triangle A D C.

00:13 A D is a median. Remember, a median divides a side into two equal parts.

00:19 So, it splits the triangle into two smaller triangles with equal areas.

00:26 The area of triangle A B C is the sum of the areas of the smaller triangles.

00:36 Plug in the given area values and solve step by step.

00:57 And that's how you find 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:

Since AD is the median

Since the area of the triangle ABC is equal to 32.

Find the area of the triangle ADC.

Step-by-step solution

To solve this problem, we'll use the property that a median of a triangle divides it into two triangles of equal area. Given the area of $\triangle ABC$ is 32:

Since $AD$ is the median, it divides $\triangle ABC$ into two triangles $\triangle ABD$ and $\triangle ADC$ of equal area.
Thus, the area of $\triangle ADC = \frac{1}{2} \times \text{Area of } \triangle ABC = \frac{1}{2} \times 32 = 16.$

Therefore, the area of triangle $\triangle ADC$ is $16$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Median Property: A median divides a triangle into two equal areas
Technique: Area of $\triangle ADC = \frac{1}{2} \times 32 = 16$
Check: Both triangles ABD and ADC have area 16, totaling 32 ✓

Common Mistakes

Avoid these frequent errors

Confusing median with altitude or angle bisector
Don't think a median affects angles or creates right triangles = wrong geometric relationships! A median only connects a vertex to the midpoint of the opposite side. Always remember that medians divide triangles into equal areas, not equal angles or perpendicular 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 a vertex to the midpoint of the opposite side. In this problem, AD connects vertex A to point D, which is the midpoint of side BC.

Why do medians always create equal areas?

Because the median divides the base into two equal parts! Since both resulting triangles share the same height from the apex, and their bases are equal, their areas must be equal too.

How is this different from an altitude?

An altitude is perpendicular to the base and creates right angles. A median goes to the midpoint but isn't necessarily perpendicular. They're completely different concepts!

What if the triangle isn't drawn to scale?

It doesn't matter! The median property works for any triangle - scalene, isosceles, or equilateral. The visual appearance doesn't change the mathematical relationship.

Can I use this property for other medians too?

Absolutely! Every median in any triangle divides it into two triangles of equal area. This is a fundamental property that always works.

How do I remember this isn't half the perimeter?

Focus on what medians do: they divide areas, not lengths. The area gets split in half because the base gets split in half, while the height stays the same.

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