Triangle ABC  is a right triangle with  AB=2 and angle BCA=30 degrees


. Find the product of the side lengths of triangle ABC 

We are given that triangle ABC is a right triangle with \( \angle BCA = 30^\circ \) and \( AB = 2 \).

In this case, since the triangle is a right triangle, \( \angle ABC = 90^\circ \) and \( \angle BAC = 60^\circ \) (because the angles in a triangle sum to \(180^\circ\)).

To solve this, we will use trigonometry to find the lengths of the other sides.

### Step 1: Label the sides
- \( AB = 2 \) (this is the hypotenuse since \( \angle ABC = 90^\circ \)).
- Let \( BC = x \) (this is the side opposite \( \angle BAC = 60^\circ \)).
- Let \( AC = y \) (this is the side adjacent to \( \angle BAC = 60^\circ \)).

### Step 2: Use trigonometric ratios
Using the sine and cosine ratios for \( \angle BAC = 60^\circ \):

\[
\sin 60^\circ = \frac{BC}{AB} = \frac{x}{2}
\]
\[
\cos 60^\circ = \frac{AC}{AB} = \frac{y}{2}
\]

### Step 3: Solve for \( x \) and \( y \)
We know that:
\[
\sin 60^\circ = \frac{\sqrt{3}}{2}, \quad \cos 60^\circ = \frac{1}{2}
\]

Using these values, we can solve for \( x \) and \( y \):

\[
\frac{\sqrt{3}}{2} = \frac{x}{2} \quad \Rightarrow \quad x = \sqrt{3}
\]
\[
\frac{1}{2} = \frac{y}{2} \quad \Rightarrow \quad y = 1
\]

### Step 4: Find the product of the side lengths
The side lengths of the triangle are \( AB = 2 \), \( BC = \sqrt{3} \), and \( AC = 1 \). The product of the side lengths is:

\[
2 \times \sqrt{3} \times 1 = 2\sqrt{3}
\]

Thus, the product of the side lengths is \( \boxed{2\sqrt{3}} \).

Would you like more details on any step? Here are some related questions to consider:

1. How can we verify the side lengths using the Pythagorean theorem?
2. What is the significance of the 30-60-90 triangle in geometry?
3. How do sine and cosine functions relate to the unit circle?
4. Can we use a different method, like area-based reasoning, to solve this problem?
5. What are the general trigonometric properties of a right triangle?

**Tip:** In a 30-60-90 triangle, the ratio of the sides is always \(1 : \sqrt{3} : 2\). This can be a quick way to solve problems involving these angles.

Triangle ABC is a right triangle with AB=2 and angle BCA=30 degrees. Find the product of the side lengths of triangle ABC.

Learn how to find the product of side lengths in a right triangle where AB=2 and angle BCA=30 degrees. This solution uses trigonometric ratios and the properties of 30-60-90 triangles, suitable for students in Grades 9-10.

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