In `DeltaABC` measure of angle B is `90^(@)`. IF `cosecA = 13//12` and AB = 10 cm, then what is the length (in cm) of sides AC ?

Options

A. 24
B. 12
C. 14
D. 26

Answer

To solve the problem step by step, we will follow the information given in the question and apply the relevant mathematical concepts.

### Step-by-Step Solution:

1. **Understanding the Triangle**:
- We have a right triangle \( \Delta ABC \) where \( \angle B = 90^\circ \).
- The sides are labeled as follows:
- \( AB = 10 \, \text{cm} \) (one leg)
- \( AC \) (hypotenuse)
- \( BC \) (other leg)

2. **Using Cosecant**:
- We are given that \( \csc A = \frac{13}{12} \).
- Recall that \( \csc A = \frac{\text{Hypotenuse}}{\text{Opposite}} \).
- Here, the hypotenuse is \( AC \) and the opposite side to angle \( A \) is \( BC \).
- Therefore, we can write:
\[ \csc A = \frac{AC}{BC} = \frac{13}{12} \]

3. **Expressing Sides in Terms of a Variable**:
- Let \( BC = 12x \) and \( AC = 13x \) for some variable \( x \).
- This gives us the relationship between the sides based on the cosecant value.

4. **Applying Pythagorean Theorem**:
- According to the Pythagorean theorem:
\[ AC^2 = AB^2 + BC^2 \]
- Substituting the known values:
\[ (13x)^2 = (10)^2 + (12x)^2 \]
- This expands to:
\[ 169x^2 = 100 + 144x^2 \]

5. **Solving for \( x \)**:
- Rearranging the equation gives:
\[ 169x^2 - 144x^2 = 100 \]
\[ 25x^2 = 100 \]
\[ x^2 = 4 \]
\[ x = 2 \]

6. **Finding the Length of Side AC**:
- Now that we have \( x \), we can find \( AC \):
\[ AC = 13x = 13 \times 2 = 26 \, \text{cm} \]

### Conclusion:
The length of side \( AC \) is \( 26 \, \text{cm} \).