`/_\ ABC` is right-angled at A. If AB = 24 cm and AC = 7 cm, then BC = ?

Options

A. 31 cm
B. 17 cm
C. 25 cm
D. 28 cm

Answer

To find the length of side BC in triangle ABC, which is right-angled at A, we will use the Pythagorean theorem. The theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

### Step-by-Step Solution:

1. **Identify the sides of the triangle**:
- In triangle ABC, angle A is the right angle (90 degrees).
- We have:
- AB (one leg) = 24 cm
- AC (the other leg) = 7 cm
- BC (the hypotenuse) = ?

2. **Apply the Pythagorean theorem**:
- According to the Pythagorean theorem:
\[ BC^2 = AB^2 + AC^2 \]

3. **Substitute the known values**:
- Substitute the lengths of AB and AC into the equation:
\[ BC^2 = (24 \, \text{cm})^2 + (7 \, \text{cm})^2 \]

4. **Calculate the squares**:
- Calculate \( AB^2 \):
\[ 24^2 = 576 \]
- Calculate \( AC^2 \):
\[ 7^2 = 49 \]

5. **Add the squares**:
- Now, add the two results:
\[ BC^2 = 576 + 49 = 625 \]

6. **Find the length of BC**:
- To find BC, take the square root of 625:
\[ BC = \sqrt{625} = 25 \, \text{cm} \]

### Final Answer:
- The length of side BC is **25 cm**.

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