In the rectangle ABCD, the diagonals intersect at point O. Find the perimeter of the AOB triangle if the CAD angle is 30 degrees, AC= 12

We are given a rectangle \( ABCD \) with diagonals intersecting at point \( O \). The diagonal \( AC = 12 \), and the angle \( \angle CAD = 30^\circ \). We need to find the perimeter of triangle \( AOB \).

### Step 1: Understanding the Geometry

- In a rectangle, the diagonals are equal and bisect each other. So, \( AC = BD \) and both diagonals intersect at \( O \), meaning \( AO = OC = 6 \).
  
- We need to find the lengths of \( AB \), \( AO \), and \( BO \) to determine the perimeter of triangle \( AOB \).

### Step 2: Using the Right Triangle \( ACD \)

Since \( \angle CAD = 30^\circ \), we can consider triangle \( ACD \), which is a right triangle (because angle \( D \) is \( 90^\circ \) in a rectangle). Using the known diagonal \( AC = 12 \), and the fact that in a 30°-60°-90° triangle, the side ratios are \( 1:\sqrt{3}:2 \), we can deduce:

- \( AD \) is half of the diagonal \( AC \), so \( AD = \frac{AC}{2} = \frac{12}{2} = 6 \).

Now, using the Pythagorean theorem for triangle \( ACD \), where \( AD = 6 \) and \( \angle ACD = 90^\circ \):

\[
AC^2 = AB^2 + AD^2
\]
\[
12^2 = AB^2 + 6^2
\]
\[
144 = AB^2 + 36
\]
\[
AB^2 = 108
\]
\[
AB = \sqrt{108} = 6\sqrt{3}
\]

### Step 3: Perimeter of Triangle \( AOB \)

- We already know \( AO = 6 \), as it is half of diagonal \( AC \).
- In a rectangle, \( AB = BO \), so \( BO = 6\sqrt{3} \).

Thus, the perimeter of triangle \( AOB \) is:

\[
P = AO + AB + BO = 6 + 6\sqrt{3} + 6\sqrt{3} = 6 + 12\sqrt{3}
\]

This is the exact value of the perimeter.

Would you like more details on any of the steps? Here are five follow-up questions to consider:

1. How do you prove that the diagonals of a rectangle are equal?
2. Can the angle \( \angle AOB \) be found based on this information?
3. How would the problem change if \( \angle CAD \) were a different value?
4. What is the area of triangle \( AOB \)?
5. How does the property of the rectangle influence the triangle \( AOB \)?

**Tip:** In any triangle, the sum of the interior angles is always \( 180^\circ \). This property is useful when solving for unknown angles.

This detailed solution explains how to find the perimeter of triangle AOB formed by diagonals in a rectangle. Learn how to use the Pythagorean theorem and side ratios for a 30°-60°-90° triangle to solve this problem. Suitable for high school students in Grades 9-12.

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