Triangle Area Ratio: Finding the Proportion when AD is a Median

Q: Why are the areas equal when the triangles look different?

Even though $ \triangle ABD $ and $ \triangle ADC $ may have different shapes, they have the same base length (BD = DC) and the same height from vertex A. Since area = $ \frac{1}{2} \times \text{base} \times \text{height} $, equal bases and heights mean equal areas!

Q: What exactly is a median in a triangle?

A median is a line segment that connects any vertex to the midpoint of the opposite side. Every triangle has exactly 3 medians, and each one divides the triangle into two smaller triangles of equal area.

Q: Does this work for any triangle shape?

Yes! Whether your triangle is scalene, isosceles, equilateral, acute, right, or obtuse, any median will always create two triangles with equal areas. This is a universal property!

Q: How can I remember this property?

Think of it like this: the median is like a fair divider that splits the triangle's area exactly in half. Since D is the midpoint of BC, it gives each smaller triangle an equal 'share' of the base, and they both reach up to the same vertex A.

Q: What if the problem asks for a different ratio like 2:3?

If AD is truly a median (meaning D is the midpoint of BC), then the ratio will always be 1:1. If you get a different ratio, double-check that D is actually the midpoint, not just any point on side BC.

Triangle Area Division with Median Lines

In front of you the next triangle:

Since AD is the median

Find the ratio of the area of the triangle ABD and the area of the triangle ADC.

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Step-by-step written solution

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Understand the problem

In front of you the next triangle:

Since AD is the median

Find the ratio of the area of the triangle ABD and the area of the triangle ADC.

Step-by-step solution

To solve this problem, we'll focus on the role of the median $AD$ .

Since $AD$ is a median, it divides side $BC$ into two equal parts, meaning $D$ is the midpoint.
The property of a median in a triangle is that it divides the triangle into two smaller triangles of equal area.
Thus, $\triangle ABD$ and $\triangle ADC$ will have the same area.

Therefore, the ratio of the area of $\triangle ABD$ to the area of $\triangle ADC$ is $1:1$ .

Final Answer

1:1

Key Points to Remember

Essential concepts to master this topic

Median Rule: A median always divides a triangle into two equal areas
Technique: Since D is midpoint of BC, triangles ABD and ADC have equal areas
Check: Both triangles share same height from A and have equal bases ✓

Common Mistakes

Avoid these frequent errors

Trying to calculate actual areas instead of using median property
Don't waste time finding base lengths and heights to calculate actual areas = unnecessary work and potential errors! The median property tells us the areas are automatically equal. Always remember that any median divides a triangle into two triangles of equal area.

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

Why are the areas equal when the triangles look different?

Even though $\triangle ABD$ and $\triangle ADC$ may have different shapes, they have the same base length (BD = DC) and the same height from vertex A. Since area = $\frac{1}{2} \times \text{base} \times \text{height}$ , equal bases and heights mean equal areas!

What exactly is a median in a triangle?

A median is a line segment that connects any vertex to the midpoint of the opposite side. Every triangle has exactly 3 medians, and each one divides the triangle into two smaller triangles of equal area.

Does this work for any triangle shape?

Yes! Whether your triangle is scalene, isosceles, equilateral, acute, right, or obtuse, any median will always create two triangles with equal areas. This is a universal property!

How can I remember this property?

Think of it like this: the median is like a fair divider that splits the triangle's area exactly in half. Since D is the midpoint of BC, it gives each smaller triangle an equal 'share' of the base, and they both reach up to the same vertex A.

What if the problem asks for a different ratio like 2:3?

If AD is truly a median (meaning D is the midpoint of BC), then the ratio will always be 1:1. If you get a different ratio, double-check that D is actually the midpoint, not just any point on side BC.

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