In `DeltaABC` measure of angle B is `90^(@)`. If `secA=25//7, and AB=14cm`, then that is the length (in cm) of side BC?

Options

A. 50
B. 48
C. 20
D. 26

Answer

To solve the problem, we need to find the length of side BC in triangle ABC, where angle B is 90 degrees, sec A is given as \( \frac{25}{7} \), and AB is 14 cm.

### Step-by-Step Solution:

1. **Understand the Triangle Configuration**:
In triangle ABC, angle B is 90 degrees, which means we can use trigonometric ratios involving angle A and angle C.

2. **Use the Secant Function**:
The secant function is defined as:
\[ \sec A = \frac{1}{\cos A} \]
Given that \( \sec A = \frac{25}{7} \), we can find \( \cos A \):
\[ \cos A = \frac{7}{25} \]

3. **Identify the Sides**:
In triangle ABC:
- AB is the side opposite to angle C.
- BC is the side opposite to angle A.
- AC is the hypotenuse.

Since \( AB = 14 \) cm, we can denote:
- \( AB = a = 14 \) cm (opposite to angle C)
- \( BC = b \) (opposite to angle A)
- \( AC = c \) (hypotenuse)

4. **Use the Cosine Definition**:
From the definition of cosine in a right triangle:
\[ \cos A = \frac{AB}{AC} \implies \frac{7}{25} = \frac{14}{c} \]
Rearranging gives:
\[ c = \frac{14 \times 25}{7} = 50 \text{ cm} \]

5. **Use the Pythagorean Theorem**:
Now we can use the Pythagorean theorem to find BC:
\[ AC^2 = AB^2 + BC^2 \implies 50^2 = 14^2 + b^2 \]
\[ 2500 = 196 + b^2 \]
\[ b^2 = 2500 - 196 = 2304 \]
\[ b = \sqrt{2304} = 48 \text{ cm} \]

6. **Conclusion**:
The length of side BC is \( 48 \) cm.

### Final Answer:
The length of side BC is **48 cm**.