In a right angled triangle ABC, the bisector of the right angle C divides AB into segment x and y and `tan((A-B)/(2))=t,` then x:y is equal to

Options

A. `(1+t) : (1-t)`
B. `(1-t) : (t+1)`
C. `1: (1+t)`
D. `(1-t) :1`

Answer

To solve the problem, we will analyze the right-angled triangle ABC with the right angle at C. The angle bisector of angle C divides the side AB into segments x and y. We are also given that \(\tan\left(\frac{A - B}{2}\right) = t\). We need to find the ratio \(x:y\).

### Step-by-Step Solution:

1. **Understanding the Triangle**:
In triangle ABC, angle C is 90 degrees. Therefore, angles A and B are complementary, meaning \(A + B = 90^\circ\).

2. **Using the Angle Bisector Theorem**:
The angle bisector of angle C divides the opposite side AB into segments x and y such that:
\[ \frac{x}{y} = \frac{a}{b} \]
where \(a\) is the length of side BC and \(b\) is the length of side AC.

3. **Relating Angles and Sides**:
We can express \(a\) and \(b\) in terms of x and y. Let:
\[ a = xk \quad \text{and} \quad b = yk \]
for some constant \(k\).

4. **Using the Given Tangent Expression**:
We know:
\[ \tan\left(\frac{A - B}{2}\right) = t \]
Using the identity for tangent of half-angle difference:
\[ \tan\left(\frac{A - B}{2}\right) = \frac{\sin(A - B)}{1 + \cos(A - B)} \]
This can also be expressed as:
\[ t = \frac{a - b}{a + b} \cdot \cot\left(\frac{C}{2}\right) \]
Since \(C = 90^\circ\), \(\cot\left(\frac{C}{2}\right) = \cot(45^\circ) = 1\).

5. **Substituting Values**:
Substituting \(a\) and \(b\):
\[ t = \frac{xk - yk}{xk + yk} = \frac{x - y}{x + y} \]
We can cancel \(k\) since it is common in both numerator and denominator.

6. **Cross Multiplying**:
Rearranging gives:
\[ xt + yt = x - y \]
Rearranging further:
\[ xt - x = -y - yt \]
Factoring out x and y:
\[ x(t - 1) = -y(1 + t) \]

7. **Finding the Ratio**:
Multiplying both sides by -1:
\[ x(1 - t) = y(1 + t) \]
Dividing both sides by y:
\[ \frac{x}{y} = \frac{1 + t}{1 - t} \]
Therefore, we can express the ratio \(x:y\) as:
\[ x:y = 1 + t : 1 - t \]

### Final Answer:
Thus, the ratio \(x:y\) is equal to \(1 + t : 1 - t\).