Let us consider a triangle ABC having BC=5 cm, CA=4cm, AB=3cm, D,E are points on BC such BD = DE= EC, `angleCAE=theta`, then:
`AE^(2)` is equal to

Options

A. `73/3`
B. `73/5`
C. `73/7`
D. `73/9`

Answer

To solve the problem, we need to find \( AE^2 \) in triangle \( ABC \) where \( BC = 5 \) cm, \( CA = 4 \) cm, and \( AB = 3 \) cm. Points \( D \) and \( E \) are on \( BC \) such that \( BD = DE = EC \), and \( \angle CAE = \theta \).

### Step-by-Step Solution:

1. **Identify the triangle and verify it is a right triangle:**
We have:
- \( AB = 3 \) cm
- \( AC = 4 \) cm
- \( BC = 5 \) cm

We can check if triangle \( ABC \) is a right triangle using the Pythagorean theorem:
\[ BC^2 = AB^2 + AC^2 \]
\[ 5^2 = 3^2 + 4^2 \implies 25 = 9 + 16 \implies 25 = 25 \]
Thus, triangle \( ABC \) is a right triangle with \( \angle A = 90^\circ \).

2. **Determine the lengths of segments \( BD \), \( DE \), and \( EC \):**
Since \( D \) and \( E \) divide \( BC \) into three equal parts:
\[ BD = DE = EC = \frac{BC}{3} = \frac{5}{3} \text{ cm} \]

3. **Use the cosine rule in triangle \( ACE \):**
We apply the cosine rule to find \( AE^2 \):
\[ AE^2 = AC^2 + CE^2 - 2 \cdot AC \cdot CE \cdot \cos(\angle CAE) \]
Here, \( AC = 4 \) cm and \( CE = \frac{5}{3} \) cm.

4. **Calculate \( \cos(\angle CAE) \):**
Since \( \angle A = 90^\circ \), we can find \( \cos C \) using triangle \( ABC \):
\[ \cos C = \frac{AC}{BC} = \frac{4}{5} \]

5. **Substituting values into the cosine rule:**
Substitute \( AC \), \( CE \), and \( \cos C \) into the cosine rule:
\[ AE^2 = 4^2 + \left(\frac{5}{3}\right)^2 - 2 \cdot 4 \cdot \frac{5}{3} \cdot \frac{4}{5} \]
\[ AE^2 = 16 + \frac{25}{9} - 2 \cdot 4 \cdot \frac{5}{3} \cdot \frac{4}{5} \]
\[ AE^2 = 16 + \frac{25}{9} - \frac{32}{3} \]

6. **Finding a common denominator:**
Convert \( 16 \) to a fraction with a denominator of \( 9 \):
\[ 16 = \frac{144}{9} \]
Convert \( \frac{32}{3} \) to a fraction with a denominator of \( 9 \):
\[ \frac{32}{3} = \frac{96}{9} \]

7. **Combine the fractions:**
Now we can combine:
\[ AE^2 = \frac{144}{9} + \frac{25}{9} - \frac{96}{9} \]
\[ AE^2 = \frac{144 + 25 - 96}{9} = \frac{73}{9} \]

### Final Result:
Thus, the value of \( AE^2 \) is:
\[ \boxed{\frac{73}{9}} \]