In an isoceles right angled triangle ABC, `angle B` is right angle, Angle bisector of `angle BAC` is AN cut at M to the median BO,. Point O lies on the hypotenuse. OM is 20 cm, then the value of AB is :

Options

A. 38.96 cm
B. 24.18 cm
C. 34.134 cm
D. none of these

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Triangle
We have an isosceles right triangle ABC where angle B is the right angle. This means that AB = BC. Let’s denote the length of AB (and BC) as \( a \).

### Step 2: Identify the Median and Angle Bisector
In triangle ABC:
- The median BO divides AC into two equal parts.
- The angle bisector AN divides angle BAC into two equal angles.

### Step 3: Determine the Length of AC
Since triangle ABC is isosceles and right-angled, we can use the Pythagorean theorem to find the length of AC:
\[ AC = \sqrt{AB^2 + BC^2} = \sqrt{a^2 + a^2} = \sqrt{2a^2} = a\sqrt{2} \]

### Step 4: Find the Length of AO
Since O is the midpoint of AC (as BO is the median), we can find AO:
\[ AO = \frac{AC}{2} = \frac{a\sqrt{2}}{2} = \frac{a}{\sqrt{2}} \]

### Step 5: Apply the Angle Bisector Theorem
According to the angle bisector theorem:
\[ \frac{AB}{AO} = \frac{BM}{MO} \]
Let \( BM = x \) and \( MO = 20 \) cm (given). Thus:
\[ \frac{a}{\frac{a}{\sqrt{2}}} = \frac{x}{20} \]
This simplifies to:
\[ \sqrt{2} = \frac{x}{20} \implies x = 20\sqrt{2} \]

### Step 6: Find the Length of BO
The length of BO can be expressed as:
\[ BO = BM + MO = 20\sqrt{2} + 20 \]

### Step 7: Calculate AB
Now, using the relation:
\[ AB = \sqrt{2} \cdot BO \]
Substituting the value of BO:
\[ AB = \sqrt{2} \cdot (20\sqrt{2} + 20) \]
This simplifies to:
\[ AB = \sqrt{2} \cdot 20\sqrt{2} + \sqrt{2} \cdot 20 = 40 + 20\sqrt{2} \]

### Step 8: Approximate the Value
Using \( \sqrt{2} \approx 1.41 \):
\[ AB \approx 40 + 20 \cdot 1.41 \approx 40 + 28.2 = 68.2 \text{ cm} \]

### Step 9: Final Calculation
The final value of AB is approximately 68.2 cm, which rounds to about 70 cm.

### Conclusion
Thus, the value of AB is approximately 70 cm.

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