In `Delta`ABC measure of angle B is `90^@`. If cosA = 5/13, and AB = 10cm, then what is the length (in cm) of side BC?

Options

A. 26
B. 24
C. 25
D. 5

Answer

To solve the problem step by step, we will use the information given about triangle ABC, where angle B is 90 degrees, cos A = 5/13, and AB = 10 cm. We need to find the length of side BC.

### Step-by-Step Solution:

1. **Identify the Triangle and Given Values**:
- We have triangle ABC with angle B = 90°.
- Given: cos A = 5/13 and AB = 10 cm.

2. **Understanding Cosine in Right Triangle**:
- In a right triangle, cos A = adjacent side / hypotenuse.
- Here, AB (adjacent to angle A) = 10 cm, and AC (hypotenuse) is what we need to find.

3. **Set Up the Equation Using Cosine**:
- From the definition of cosine:
\[ \cos A = \frac{AB}{AC} \]
- Substitute the known values:
\[ \frac{5}{13} = \frac{10}{AC} \]

4. **Cross-Multiply to Solve for AC**:
- Cross-multiplying gives:
\[ 5 \cdot AC = 10 \cdot 13 \]
- Simplifying this:
\[ 5 \cdot AC = 130 \]
- Now, divide both sides by 5:
\[ AC = \frac{130}{5} = 26 \text{ cm} \]

5. **Use Pythagorean Theorem to Find BC**:
- In triangle ABC, we can use the Pythagorean theorem:
\[ AC^2 = AB^2 + BC^2 \]
- Substitute the known lengths:
\[ 26^2 = 10^2 + BC^2 \]
- Calculate the squares:
\[ 676 = 100 + BC^2 \]

6. **Isolate BC^2**:
- Rearranging gives:
\[ BC^2 = 676 - 100 = 576 \]

7. **Calculate BC**:
- Taking the square root:
\[ BC = \sqrt{576} = 24 \text{ cm} \]

### Final Answer:
The length of side BC is **24 cm**.