https://file.aiquickdraw.com/mathbot/file/1726406910737u70rnaed.jpg

The given image presents a rectangle \( ABCD \) with an additional diagonal \( BD \). The angles and side relationships are labeled as follows:

- \( \angle DAB = 27^\circ \)
- \( AB = 3x \)
- \( BD = y \)

The task asks to find the values of \( x \) and \( y \).

### Step-by-step solution:

Since \( ABCD \) is a rectangle, we know that the diagonals divide the rectangle into two right triangles. In the triangle \( ABD \), the angle \( \angle DAB = 27^\circ \), and we can use trigonometry to express the relationships between the sides.

1. **Using Trigonometry:**

   In right triangle \( ABD \), we can use the trigonometric functions:
   
   - \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \)
   - \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \)
   
   Here, \( AB = 3x \), \( AD = x \), and \( BD = y \).
   
   Using \( \sin(27^\circ) \) and \( \cos(27^\circ) \), we set up the equations:

   \[
   \sin(27^\circ) = \frac{AB}{BD} = \frac{3x}{y}
   \]
   \[
   \cos(27^\circ) = \frac{AD}{BD} = \frac{x}{y}
   \]

2. **Solving for \( x \) and \( y \):**

   Rearranging the first equation:

   \[
   y = \frac{3x}{\sin(27^\circ)}
   \]

   Rearranging the second equation:

   \[
   y = \frac{x}{\cos(27^\circ)}
   \]

   Now we equate both expressions for \( y \):

   \[
   \frac{3x}{\sin(27^\circ)} = \frac{x}{\cos(27^\circ)}
   \]

   Simplifying the equation by dividing both sides by \( x \) (assuming \( x \neq 0 \)):

   \[
   \frac{3}{\sin(27^\circ)} = \frac{1}{\cos(27^\circ)}
   \]

   Cross-multiplying gives:

   \[
   3

Calculate the values of x and y for the given rectangle ABCD, with AB = 3x, BD = y, and angle DAB = 27°.

Solve the trigonometric problem involving rectangle ABCD with a diagonal BD and an angle of 27°. Learn how to calculate the values of x and y using sine and cosine relationships in right triangles. Suitable for students in Grades 10-12.

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