in the figure above, ABCD is a square. M is the point one-third of the way from B to C. N is the point one-half of the way from D to C. Then `theta`=

Options

A. `50.8^(@)`
B. `45.0^(@)`
C. `36.9^(@)`
D. `36.1^(@)`

Explanation

To solve the problem step-by-step, we will analyze the square ABCD and the points M and N as described in the question.

### Step 1: Define the square and points
Let the side length of square ABCD be \( x \). The coordinates of the points can be defined as follows:
- \( A(0, 0) \)
- \( B(0, x) \)
- \( C(x, x) \)
- \( D(x, 0) \)

### Step 2: Determine the coordinates of points M and N
- Point M is one-third of the way from B to C. Therefore, the coordinates of M can be calculated as:
\[ M = B + \frac{1}{3}(C - B) = \left(0, x\right) + \frac{1}{3}\left((x, x) - (0, x)\right) = \left(0, x\right) + \frac{1}{3}(x, 0) = \left(\frac{x}{3}, x\right) \]

- Point N is one-half of the way from D to C. Therefore, the coordinates of N can be calculated as:
\[ N = D + \frac{1}{2}(C - D) = \left(x, 0\right) + \frac{1}{2}\left((x, x) - (x, 0)\right) = \left(x, 0\right) + \frac{1}{2}(0, x) = \left(x, \frac{x}{2}\right) \]

### Step 3: Calculate angles BAM and DAN
- For angle BAM:
- The length \( BM = \frac{x}{3} \) (vertical distance from B to M).
- The length \( AB = x \) (horizontal distance from A to B).

The tangent of angle BAM is given by:
\[ \tan(\angle BAM) = \frac{BM}{AB} = \frac{\frac{x}{3}}{x} = \frac{1}{3} \]

Therefore, \( \angle BAM = \tan^{-1}\left(\frac{1}{3}\right) \).

- For angle DAN:
- The length \( DN = \frac{x}{2} \) (vertical distance from D to N).
- The length \( AD = x \) (horizontal distance from A to D).

The tangent of angle DAN is given by:
\[ \tan(\angle DAN) = \frac{DN}{AD} = \frac{\frac{x}{2}}{x} = \frac{1}{2} \]

Therefore, \( \angle DAN = \tan^{-1}\left(\frac{1}{2}\right) \).

### Step 4: Use the angles to find theta
Since angle \( DAB \) is a right angle (90 degrees), we can express it as:
\[ \angle DAB = \angle DAN + \theta + \angle BAM \]
Substituting the known values:
\[ 90^\circ = \tan^{-1}\left(\frac{1}{2}\right) + \theta + \tan^{-1}\left(\frac{1}{3}\right) \]

### Step 5: Calculate the angles
Using approximate values:
- \( \tan^{-1}\left(\frac{1}{3}\right) \approx 18.4^\circ \)
- \( \tan^{-1}\left(\frac{1}{2}\right) \approx 26.6^\circ \)

Substituting these values into the equation:
\[ 90^\circ = 26.6^\circ + \theta + 18.4^\circ \]
\[ 90^\circ = 45^\circ + \theta \]
\[ \theta = 90^\circ - 45^\circ = 45^\circ \]

### Final Answer
Thus, the value of \( \theta \) is \( 45^\circ \).