Two medians AD and BE of `Delta ABC` intersect at G at right angles . If `AD=9 cm and BE=6 cm`, them the length of BD (in cm) is

Answer

To solve the problem, we need to find the length of segment BD in triangle ABC, where two medians AD and BE intersect at point G at right angles. We are given that AD = 9 cm and BE = 6 cm.

### Step-by-Step Solution:

1. **Understand the Problem**: We have triangle ABC with medians AD and BE intersecting at point G. We know the lengths of the medians: AD = 9 cm and BE = 6 cm. We need to find the length of segment BD.

2. **Draw the Triangle**: Draw triangle ABC with points A, B, and C. Draw median AD from vertex A to the midpoint D of side BC, and median BE from vertex B to the midpoint E of side AC. Mark the intersection point of the medians as G.

3. **Identify the Ratios**: The centroid G divides each median in a 2:1 ratio. This means:
- For median AD: AG:GD = 2:1
- For median BE: BG:GE = 2:1

4. **Calculate AG and GD**: Since AD = 9 cm, we can express AG and GD in terms of a variable x:
- AG + GD = AD
- Let AG = 2x and GD = x, then:
\[ 2x + x = 9 \implies 3x = 9 \implies x = 3 \]
- Therefore, AG = 2x = 6 cm and GD = x = 3 cm.

5. **Calculate BG and GE**: Since BE = 6 cm, we can express BG and GE in terms of another variable y:
- BG + GE = BE
- Let BG = 2y and GE = y, then:
\[ 2y + y = 6 \implies 3y = 6 \implies y = 2 \]
- Therefore, BG = 2y = 4 cm and GE = y = 2 cm.

6. **Apply the Pythagorean Theorem**: In triangle BGD, we can use the Pythagorean theorem since G is a right angle:
\[ BD^2 = BG^2 + GD^2 \]
Substituting the known values:
\[ BD^2 = (4 \text{ cm})^2 + (3 \text{ cm})^2 \]
\[ BD^2 = 16 + 9 = 25 \]
\[ BD = \sqrt{25} = 5 \text{ cm} \]

7. **Conclusion**: The length of segment BD is 5 cm.