In `DeltaABC`, the median AD is 7 cm and CB is 14 cm, measure of `/_CAB `is

Options

A. `30^@`
B. `60^@`
C. `90^@`
D. `120^@`

Answer

To solve the problem, we need to find the measure of angle CAB in triangle ABC, given that the median AD is 7 cm and side CB is 14 cm.

### Step-by-step Solution:

1. **Understanding the Triangle and Median**:
- In triangle ABC, AD is the median. This means that point D is the midpoint of side BC.
- Since CB = 14 cm, and D is the midpoint, we can conclude that BD = DC = 7 cm (because the median divides the opposite side into two equal parts).

2. **Identifying the Triangle**:
- Now we have triangle ABD where:
- AD = 7 cm (the median)
- BD = 7 cm (half of CB)
- This gives us an isosceles triangle ABD, where AB = AD and BD = 7 cm.

3. **Using the Properties of Isosceles Triangle**:
- In isosceles triangle ABD, since AD = BD, the angles opposite to these sides must be equal. Therefore, angle ADB = angle ABD.

4. **Applying the Pythagorean Theorem**:
- Since AD is the median and also equal to the length of BD, we can apply the Pythagorean theorem in triangle ABD to find angle CAB.
- In triangle ABD:
- AB² = AD² + BD²
- AB² = 7² + 7²
- AB² = 49 + 49
- AB² = 98
- AB = √98 = 7√2 cm.

5. **Finding the Angle CAB**:
- Now, we can find angle CAB using the cosine rule or recognizing that triangle ABD is an isosceles right triangle.
- Since AD = BD, angle ADB = angle ABD = 45 degrees (as the angles in a triangle sum up to 180 degrees).
- Therefore, angle CAB = 90 degrees.

### Conclusion:
The measure of angle CAB is 90 degrees.