Let ABC and AB'C be two non-congruent triangles with sides BC=B'C=5, AC=6, and `angleA` is fixed. If `A_(1)` and `A_(2)` are the area of the two triangles ABC and AB'C, then the value of `(A_(1)^(2)+A_(2)^(2)-2A_(1)A_(2)cos 2A)/((A_(1)+A_(2))^(2))` is

Options

A. `9//36`
B. `25//36`
C. `25//16`
D. `16//25`

Explanation

To solve the problem, we need to find the value of the expression

\[ \frac{A_1^2 + A_2^2 - 2A_1A_2 \cos 2A}{(A_1 + A_2)^2} \]

where \(A_1\) and \(A_2\) are the areas of triangles \(ABC\) and \(AB'C\) respectively, with given conditions.

### Step 1: Determine the areas \(A_1\) and \(A_2\)

The area of a triangle can be calculated using the formula:

\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]

For triangles \(ABC\) and \(AB'C\), we can express the areas in terms of the sides and the sine of angle \(A\):

\[ A_1 = \frac{1}{2} \times BC \times h_1 = \frac{1}{2} \times 5 \times h_1 \]
\[ A_2 = \frac{1}{2} \times B'C \times h_2 = \frac{1}{2} \times 5 \times h_2 \]

where \(h_1\) and \(h_2\) are the heights from point \(A\) to line \(BC\) and \(B'C\) respectively.

### Step 2: Express heights in terms of angle \(A\)

Using the sine function, we can express the heights \(h_1\) and \(h_2\):

\[ h_1 = AC \cdot \sin A = 6 \cdot \sin A \]
\[ h_2 = AC \cdot \sin A = 6 \cdot \sin A \]

Thus, we have:

\[ A_1 = \frac{1}{2} \times 5 \times (6 \sin A) = 15 \sin A \]
\[ A_2 = \frac{1}{2} \times 5 \times (6 \sin A) = 15 \sin A \]

### Step 3: Calculate \(A_1^2\) and \(A_2^2\)

Since both areas are equal, we have:

\[ A_1 = A_2 = 15 \sin A \]

Thus,

\[ A_1^2 = (15 \sin A)^2 = 225 \sin^2 A \]
\[ A_2^2 = (15 \sin A)^2 = 225 \sin^2 A \]

### Step 4: Substitute into the expression

Now we substitute \(A_1\) and \(A_2\) into the expression:

\[ A_1^2 + A_2^2 = 225 \sin^2 A + 225 \sin^2 A = 450 \sin^2 A \]

Next, we calculate \(2A_1A_2 \cos 2A\):

\[ 2A_1A_2 = 2 \times (15 \sin A) \times (15 \sin A) = 450 \sin^2 A \]

Thus,

\[ 2A_1A_2 \cos 2A = 450 \sin^2 A \cos 2A \]

### Step 5: Calculate the denominator

Now, we calculate \((A_1 + A_2)^2\):

\[ A_1 + A_2 = 15 \sin A + 15 \sin A = 30 \sin A \]
\[ (A_1 + A_2)^2 = (30 \sin A)^2 = 900 \sin^2 A \]

### Step 6: Substitute everything into the expression

Now we substitute everything into the original expression:

\[ \frac{450 \sin^2 A - 450 \sin^2 A \cos 2A}{900 \sin^2 A} \]

### Step 7: Simplify the expression

Factor out \(450 \sin^2 A\):

\[ = \frac{450 \sin^2 A (1 - \cos 2A)}{900 \sin^2 A} \]

This simplifies to:

\[ = \frac{450 (1 - \cos 2A)}{900} = \frac{1 - \cos 2A}{2} \]

Using the identity \(1 - \cos 2A = 2 \sin^2 A\), we get:

\[ = \frac{2 \sin^2 A}{2} = \sin^2 A \]

### Final Result

Thus, the value of the expression is:

\[ \sin^2 A \]