G is the centroid of `Delta ABC`. The medians AD and BE intersect at right angles . If the lengths of AD and BE are 9 cm and 12 cm respectively , then the length of AB (in cm ) is ?

Answer

To solve the problem, we need to find the length of side AB in triangle ABC, given that G is the centroid, and the medians AD and BE intersect at right angles. The lengths of the medians are AD = 9 cm and BE = 12 cm.

### Step-by-Step Solution:

1. **Understanding the Centroid and Medians**:
- The centroid (G) of a triangle divides each median into a ratio of 2:1. This means that if we denote AG as 2x and GD as x, then AD = AG + GD = 3x.
- Similarly, for median BE, we can denote BG as 2y and GE as y, giving us BE = BG + GE = 3y.

2. **Setting Up the Equations**:
- From the problem, we know:
- AD = 9 cm → 3x = 9 → x = 3 cm
- BE = 12 cm → 3y = 12 → y = 4 cm

3. **Finding AG and BG**:
- Now we can find the lengths of AG and BG:
- AG = 2x = 2(3) = 6 cm
- BG = 2y = 2(4) = 8 cm

4. **Applying the Pythagorean Theorem**:
- Since the medians intersect at right angles, we can apply the Pythagorean theorem in triangle AGB:
- AB² = AG² + BG²
- AB² = (6 cm)² + (8 cm)²
- AB² = 36 + 64
- AB² = 100

5. **Calculating AB**:
- Taking the square root to find AB:
- AB = √100 = 10 cm

### Conclusion:
The length of AB is **10 cm**.