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The provided image shows a right triangle with the following labeled information:

- One of the angles is \(30^\circ\), and another is \(60^\circ\), with the right angle being \(90^\circ\).
- The hypotenuse is labeled as 17.
- The sides opposite and adjacent to the \(30^\circ\) and \(60^\circ\) angles are labeled \(a\), \(b\), and \(c\) accordingly.

We need to find the exact values for the sides labeled \(a\), \(b\), and \(c\). We can use the trigonometric properties of a 30-60-90 triangle.

### Step-by-step solution:
1. In a 30-60-90 triangle, the sides are in the ratio:
   \[
   1 : \sqrt{3} : 2
   \]
   Specifically:
   - The side opposite the \(30^\circ\) angle (side \(b\)) is half the hypotenuse.
   - The side opposite the \(60^\circ\) angle (side \(a\)) is \( \sqrt{3} \) times the side opposite the \(30^\circ\) angle.
   - The hypotenuse is the longest side.

2. Given that the hypotenuse \(c = 17\), we can now calculate the sides:

   - Side \(b\) (opposite the \(30^\circ\) angle):
     \[
     b = \frac{1}{2} \times 17 = 8.5
     \]

   - Side \(a\) (opposite the \(60^\circ\) angle):
     \[
     a = b \times \sqrt{3} = 8.5 \times \sqrt{3} = 8.5 \times 1.732 = 14.722
     \]

   So, the exact values are:
   - \( a = 8.5 \times \sqrt{3} \approx 14.722 \)
   - \( b = 8.5 \)
   - \( c = 17 \)

Let me know if you would like a more detailed explanation or further assistance!

---

Would you like additional details or clarifications on this? Here are some related questions:

1. How are the trigonometric ratios derived for special right triangles?
2. What is the significance of the 30-60-90 triangle's side ratios in geometry?
3. How do you solve for unknown sides in a right triangle using sine, cosine, and tangent functions?
4. What are the differences between a 45-45-90 triangle and a 30-60-90 triangle?
5. How can you verify the side lengths of a right triangle using the Pythagorean Theorem?

**Tip:** Always remember the side ratios for special triangles like the 30-60-90 and 45-45-90 triangles; they help solve problems quickly!

Find the exact value of each part labeled with a variable in the following figure. The triangle has a hypotenuse of 17, with angles of 30°, 60°, and 90°, and sides labeled a, b, and c.

This solution provides a step-by-step guide to finding the sides of a 30°-60°-90° triangle with a hypotenuse of 17. Using trigonometric ratios and properties of special triangles, the sides a, b, and c are calculated exactly. Suitable for high school math students in Grades 9-12.

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