Triangle Area Calculation: Using Median AD to Find Area of Triangle ABC from Area 15

Q: What exactly is a median in a triangle?

A median is a line segment from any vertex to the midpoint of the opposite side. In this case, AD goes from vertex A to point D, which is the midpoint of side BC.

Q: Does this work for any triangle shape?

Yes! Whether the triangle is acute, obtuse, or right-angled, any median will always divide it into two triangles of equal area. This is a fundamental property.

Median Properties with Equal Area Division

In front of you the next triangle:

Since AD is the median

Since the area of the triangle ADB is equal to 15.

Find the area of the triangle ABC.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Determine the area of triangle ABC

00:03 AD is a median according to the given data, a median bisects the side

00:17 The median to a side in a triangle creates two triangles of equal area

00:24 Substitute in the area value according to the given data

00:31 The area of triangle ABC equals the sum of the areas of the triangles within it

00:43 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:

Since AD is the median

Since the area of the triangle ADB is equal to 15.

Find the area of the triangle ABC.

Step-by-step solution

To solve this problem, we'll employ the theorem which states that the median of a triangle divides it into two triangles of equal area. Given that $AD$ is a median of $\triangle ABC$ , it divides $\triangle ABC$ into $\triangle ADB$ and $\triangle ADC$ .

Step 1: Recognize the properties of a median. The median $AD$ implies:
- Area of $\triangle ADB =$ Area of $\triangle ADC$ .
- Given Area of $\triangle ADB = 15$ , hence Area of $\triangle ADC = 15$ .

Step 2: Compute the total area of $\triangle ABC$ :
- Total Area of $\triangle ABC =$ Area of $\triangle ADB +$ Area of $\triangle ADC = 15 + 15 = 30.$

Thus, the area of triangle $\triangle ABC$ is $\boxed{30}$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Median Rule: A median divides any triangle into two equal areas
Technique: If triangle ADB has area 15, then triangle ADC also has area 15
Check: Total area ABC = 15 + 15 = 30, verify both parts are equal ✓

Common Mistakes

Avoid these frequent errors

Assuming the median creates unequal areas
Don't think triangle ADB and ADC have different areas = wrong total calculation! A median always creates two triangles with identical areas, regardless of triangle shape. Always remember that medians divide triangles into equal halves.

Practice Quiz

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Is DE side in one of the triangles?

FAQ

Everything you need to know about this question

What exactly is a median in a triangle?

A median is a line segment from any vertex to the midpoint of the opposite side. In this case, AD goes from vertex A to point D, which is the midpoint of side BC.

Why does a median always create equal areas?

Because the median splits the base in half! Both triangles ADB and ADC share the same height from A, but each has half the base (BD = DC), so their areas must be equal.

Does this work for any triangle shape?

Yes! Whether the triangle is acute, obtuse, or right-angled, any median will always divide it into two triangles of equal area. This is a fundamental property.

What if I'm given the area of triangle ADC instead?

The process is identical! If triangle ADC has area 15, then triangle ADB also has area 15, making the total area of triangle ABC equal to 30.

Can I use this property to find unknown areas?

If you know the total area of triangle ABC, divide by 2 to get each part
If you know one part's area, double it to get the total
This works with any median in any triangle!

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