Assertion (A): x = $5\sqrt{2}$

Reason (R): AC² = 8² + 6² = x² + x²

1. A is true, but R is false.

2. A is false, but R is true.

3. Both A and R are true, and R is the correct reason for A.

4. Both A and R are true, and R is the incorrect reason for A.

In Δ ABC,

Since angle ABC = 90°.

According to Pythagoras theorem, in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

⇒ Hypotenuse² = Base² + Height²

⇒ AC² = AB² + BC²

⇒ AC² = 8² + 6² ……………………(1)

In Δ ADC,

Since angle ADC = 90°.

Using pythagoras theorem,

⇒ AC² = AD² + DC²

⇒ AC² = x² + x² …………………(2)

From equation (1) and (2),

⇒ 8² + 6² = x² + x²

⇒ 64 + 36 = 2x²

⇒ 2x² = 100

⇒ x² = $\dfrac{100}{2}$

⇒ x² = 50

⇒ x = $\sqrt{50}$

⇒ x = 5 $\sqrt{2}$

∴ Both A and R are true, and R is the correct reason for A.

Hence, option 3 is the correct option.