In a triangle ABC , let E be the midpoint of AC and F be the midpoint of AB .

The medians BE and CF intersect at G . Let Y and Z be the midpoints of BE

and CF respectively. If the area of triangle ABC is 480 , find the area of triangle

GY Z.

In this problem, we are tasked with finding the area of triangle \( GYZ \), given the area of triangle \( ABC \) as 480. Let's break down the solution step by step.

### Step 1: Understanding the Medians and Centroid
- **E** is the midpoint of side \( AC \), and **F** is the midpoint of side \( AB \). This means \( BE \) and \( CF \) are medians.
- The point **G** is the centroid of triangle \( ABC \), which is the intersection of the medians \( BE \) and \( CF \).
  
The key property of the centroid \( G \) is that it divides each median into a ratio of \( 2:1 \), with the longer segment being closer to the vertex. This means the centroid divides each median into one part that is two-thirds the length of the median and another part that is one-third the length.

### Step 2: Midpoints \( Y \) and \( Z \)
- **Y** is the midpoint of the median \( BE \), and **Z** is the midpoint of the median \( CF \).

Since both \( Y \) and \( Z \) are midpoints of these medians, each line segment from \( G \) to \( Y \) and from \( G \) to \( Z \) is half the length of the corresponding part of the median.

### Step 3: Area Relation in a Triangle
The centroid \( G \) divides triangle \( ABC \) into six smaller triangles of equal area. This is because the medians divide the triangle into six smaller triangles, all of which have the same area. Therefore, each of these smaller triangles has an area of:

\[
\text{Area of each smaller triangle} = \frac{480}{6} = 80
\]

### Step 4: Area of Triangle \( GYZ \)
The triangle \( GYZ \) is formed by joining the midpoints \( Y \) and \( Z \) with the centroid \( G \). Since \( Y \) and \( Z \) are midpoints, triangle \( GYZ \) lies within one of the smaller triangles created by the medians. Further, it is known that triangle \( GYZ \) occupies one-fourth of the area of the smaller triangle.

Thus, the area of triangle \( GYZ \) is:

\[
\text{Area of } GYZ = \frac{1}{4} \times 80 = 20
\]

### Final Answer:
The area of triangle \( GYZ \) is \( \boxed{20} \).

Would you like more details or have any further questions?

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Here are 5 questions to expand the discussion:
1. How does the centroid divide the medians in a triangle?
2. Can you prove why the centroid divides a triangle into six equal-area triangles?
3. What happens to the area if one of the midpoints is moved away from the centroid?
4. How would the area of triangle \( GYZ \) change if triangle \( ABC \) were not equilateral?
5. How can we generalize this problem to triangles with arbitrary areas?

**Tip:** Always remember that the centroid divides a triangle into six smaller triangles of equal area!

In a triangle ABC, let E be the midpoint of AC and F be the midpoint of AB. The medians BE and CF intersect at G. Let Y and Z be the midpoints of BE and CF respectively. If the area of triangle ABC is 480, find the area of triangle GYZ.

Learn how to calculate the area of triangle GYZ, formed by the centroid and medians of triangle ABC, with a step-by-step guide. This geometry problem explores the centroid's properties, medians, and area calculations, ideal for students in Grades 9-11. Given the area of triangle ABC as 480, this solution simplifies the process of finding the area of triangle GYZ as 20.

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