G is the centroid of `DeltaABC`. The medians AD and BE Intersect at right angles. If the lengths of AD and BE are 9 cm and 12 cm re spectively, then the length of AB (in cm) is

Options

A. `9.5 `
B. `10`
C. `11`
D. `10.5`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Given Information
We have a triangle ABC with G as the centroid. The lengths of the medians AD and BE are given as:
- AD = 9 cm
- BE = 12 cm

### Step 2: Determine the Lengths of AG, GD, BG, and GE
The centroid divides each median in a ratio of 2:1. Therefore, we can express the lengths of AG and GD as follows:
- Let AG = 2x and GD = x. Then, AD = AG + GD = 2x + x = 3x.
- Since AD = 9 cm, we have:
\[ 3x = 9 \implies x = 3 \implies AG = 2x = 6 \text{ cm}, \quad GD = x = 3 \text{ cm} \]

Next, for median BE:
- Let BG = 2y and GE = y. Then, BE = BG + GE = 2y + y = 3y.
- Since BE = 12 cm, we have:
\[ 3y = 12 \implies y = 4 \implies BG = 2y = 8 \text{ cm}, \quad GE = y = 4 \text{ cm} \]

### Step 3: Analyze Triangle AGB
Now, we focus on triangle AGB, where:
- AG = 6 cm
- BG = 8 cm
- Angle AGB = 90 degrees (since the medians intersect at right angles)

### Step 4: Apply the Pythagorean Theorem
To find the length of AB, we can use the Pythagorean theorem:
\[ AB^2 = AG^2 + BG^2 \]
Substituting the values we found:
\[ AB^2 = 6^2 + 8^2 = 36 + 64 = 100 \]
Taking the square root:
\[ AB = \sqrt{100} = 10 \text{ cm} \]

### Step 5: Conclusion
The length of AB is 10 cm.

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