Which of the following is true in the given figure, where AD is the altitude to the hypotenuse of a right-angled ΔABC?

(I) ΔABD ∼ ΔCAD

(II) ΔADB ≅ ΔCDA

(III) ΔADB ∼ ΔCAB

1. I and II

2. II and III

3. I and III

4. I, II and III

In ΔABD,

∠B + ∠BAD = 90°

∠BAD = 90° - ∠B ……..(1)

In ΔABC,

Since, BC is the hypotenuse, thus angle A = 90°

From figure,

∠DAC = ∠BAC - ∠BAD

∠DAC = 90° - ∠BAD

Substituting value of ∠BAD from equation (1) in above equation, we get :

∠DAC = 90° - (90° - ∠B) = ∠B.

In ΔABD and ΔCAD,

∠DAC = ∠B (Proved above)

∠ADB = ∠ADC (Both equal to 90°)

∴ ΔABD ∼ ΔCAD by AA similarity.

Thus, (I) is true.

or we can say that ΔADB ∼ ΔCDA.

Thus, (II) is true.

In ΔADB and ΔCAB,

∠ADB = ∠CAB = 90°

∠ABD = ∠CBA [Common angle]

∴ ΔADB ∼ ΔCAB by AA similarity.

Thus, (III) is true.

Hence, option 4 is the correct option.