`/_\ ABC` is an isosceles triangle with `/_C = 90^(@)` and AC = 5 cm. Then, AB = ?

Options

A. 2.5 cm
B. 5 cm
C. 10 cm
D. `5sqrt2 cm`

Answer

To find the length of AB in the isosceles triangle ABC where angle C is 90 degrees and AC = 5 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Triangle Properties**:
- Triangle ABC is an isosceles triangle with angle C = 90 degrees.
- Since it is isosceles and angle C is the right angle, we have AC = BC.

2. **Assign Lengths**:
- Given that AC = 5 cm, we can also conclude that BC = 5 cm (since AC = BC in an isosceles triangle).

3. **Apply the Pythagorean Theorem**:
- In a right triangle, the Pythagorean theorem states that:
\[ AB^2 = AC^2 + BC^2 \]
- Substituting the known values:
\[ AB^2 = 5^2 + 5^2 \]

4. **Calculate the Squares**:
- Calculate \(5^2\):
\[ 5^2 = 25 \]
- Therefore:
\[ AB^2 = 25 + 25 = 50 \]

5. **Find the Length of AB**:
- To find AB, take the square root of both sides:
\[ AB = \sqrt{50} \]
- Simplifying \(\sqrt{50}\):
\[ AB = \sqrt{25 \times 2} = 5\sqrt{2} \text{ cm} \]

### Final Answer:
AB = \(5\sqrt{2}\) cm

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