ABCD is a parallelogram. E & F are centroid of triangles ABD and BDC respectively. Find the length of EF if length of diagonal AC = 12 cm.

Answer

To find the length of EF in the parallelogram ABCD, where E and F are the centroids of triangles ABD and BDC respectively, and the length of diagonal AC is given as 12 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Properties of the Parallelogram:**
In a parallelogram, the diagonals bisect each other. Therefore, if AC is the diagonal, then the midpoint O of AC will divide it into two equal parts. Since AC = 12 cm, AO = OC = 6 cm.

**Hint:** Remember that in a parallelogram, the diagonals bisect each other.

2. **Identifying the Centroids:**
The centroid of a triangle divides each median into a ratio of 2:1. For triangle ABD, E is the centroid, and for triangle BDC, F is the centroid.

3. **Finding the Length of AE:**
The median AD in triangle ABD connects vertex A to the midpoint of BD. Since E is the centroid, it divides the median AD in the ratio 2:1. Therefore, if we let the length of AD be represented as 6 cm (the same as AO), then AE = (2/3) * AD = (2/3) * 6 cm = 4 cm.

**Hint:** The centroid divides the median into two segments with a ratio of 2:1.

4. **Finding the Length of BF:**
Similarly, for triangle BDC, F is the centroid. The median from B to the midpoint of DC will also be divided in the same ratio. Therefore, BF = (2/3) * BD. Since BD is also equal to 6 cm, we find BF = (2/3) * 6 cm = 4 cm.

**Hint:** Use the same ratio for the centroid in triangle BDC.

5. **Calculating the Length of EF:**
Since both E and F are located along the line connecting the midpoints of the sides of the parallelogram, the distance EF can be calculated as the sum of AE and BF. Thus, EF = AE + BF = 4 cm + 4 cm = 8 cm.

**Hint:** The total length EF is the sum of the distances from the centroids to the midpoint of the diagonal.

### Final Answer:
The length of EF is 8 cm.