In `DeltaABC, AB = a - b, AC = sqrt(a^2 + b^2)` and `BC = sqrt(2ab)`, then find angle B.

Options

A. `60^@`
B. `30^@`
C. `90^@`
D. `45^@`

Answer

To find angle B in triangle ABC where the sides are given as \( AB = a - b \), \( AC = \sqrt{a^2 + b^2} \), and \( BC = \sqrt{2ab} \), we will use the Pythagorean theorem.

### Step-by-Step Solution:

1. **Identify the sides of the triangle:**
- Let \( AB = c = a - b \)
- Let \( AC = b = \sqrt{a^2 + b^2} \)
- Let \( BC = a = \sqrt{2ab} \)

2. **Apply the Pythagorean theorem:**
According to the Pythagorean theorem, in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. We will check if:
\[ AB^2 + BC^2 = AC^2 \]
Substituting the values:
\[ (a - b)^2 + (\sqrt{2ab})^2 = (\sqrt{a^2 + b^2})^2 \]

3. **Expand the left-hand side:**
- Calculate \( (a - b)^2 \):
\[ (a - b)^2 = a^2 - 2ab + b^2 \]
- Calculate \( (\sqrt{2ab})^2 \):
\[ (\sqrt{2ab})^2 = 2ab \]
- Now combine these:
\[ a^2 - 2ab + b^2 + 2ab = a^2 + b^2 \]

4. **Simplify the left-hand side:**
\[ a^2 + b^2 = a^2 + b^2 \]
This shows that the left-hand side equals the right-hand side.

5. **Conclusion:**
Since the equation holds true, by the Pythagorean theorem, triangle ABC is a right triangle with angle B being \( 90^\circ \).

Thus, the measure of angle B is \( 90^\circ \).