Which of the following is true, in the given figure, where AD is the altitude to the hypotenuse of a right angled `triangle ABC`?
`triangle CAD` and `triangle ABD` are similar
`triangle CDA` and `triangle ADB` are congruent
`triangle ADB` and `triangle CAB` are similar
Select the correct answer using the codes given below :

Options

A. A)1 and 2
B. B)2 and 3
C. C)1 and 3
D. D)1, 2 and 3

Answer

To solve the problem, we need to analyze the relationships between the triangles formed when an altitude is drawn from vertex A to the hypotenuse BC in right triangle ABC.

### Step-by-Step Solution:

1. **Identify the Triangles**:
- We have triangle ABC, where angle A is the right angle (90 degrees).
- The altitude AD is drawn from vertex A to the hypotenuse BC, creating two smaller triangles: triangle ABD and triangle ACD.

2. **Determine Angles in Triangle ABD**:
- In triangle ABD:
- Angle ADB = 90 degrees (since AD is the altitude).
- Let angle CAB = θ (this is one of the angles in triangle ABC).
- Therefore, angle ABD = 90 - θ = α (the other angle in triangle ABC).

3. **Determine Angles in Triangle ACD**:
- In triangle ACD:
- Angle ADC = 90 degrees (since AD is the altitude).
- Angle ACD = θ (same as angle CAB).
- Therefore, angle CAD = 90 - θ = α.

4. **Check Similarity of Triangle CAD and Triangle ABD**:
- Triangle CAD has angles: θ, α, and 90 degrees.
- Triangle ABD has angles: θ, α, and 90 degrees.
- Since both triangles have the same angles, they are similar by AA (Angle-Angle) similarity criterion.

5. **Check Congruence of Triangle CDA and Triangle ADB**:
- Triangle CDA has angles: θ, 90 degrees, and α.
- Triangle ADB has angles: θ, 90 degrees, and α.
- While they are similar, we cannot conclude they are congruent unless their corresponding sides are equal. Therefore, this statement is false.

6. **Check Similarity of Triangle ADB and Triangle CAB**:
- Triangle ADB has angles: θ, α, and 90 degrees.
- Triangle CAB has angles: θ, α, and 90 degrees.
- Since both triangles have the same angles, they are also similar by AA similarity criterion.

### Conclusion:
- The correct statements are:
- Triangle CAD and triangle ABD are similar (True).
- Triangle CDA and triangle ADB are congruent (False).
- Triangle ADB and triangle CAB are similar (True).

Thus, the correct answer is options 1 and 3.