Determine if the following lengths make an acute, right or obtuse triangle. 7. 15, 16, 17.

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To determine whether the triangle with side lengths **15, 16, 17** is acute, right, or obtuse, we can use the **Pythagorean Theorem** as a guideline. Here's how you can analyze it:

1. **Arrange the Sides:**
   - Let $$ a = 15 $$, $$ b = 16 $$, and $$ c = 17 $$ (where $$ c $$ is the longest side).

2. **Apply the Pythagorean Theorem:**
   - **Calculate $$ a^2 + b^2 $$:**
     $$
     15^2 + 16^2 = 225 + 256 = 481
     $$
   - **Calculate $$ c^2 $$:**
     $$
     17^2 = 289
     $$

3. **Compare the Sums:**
   - If $$ a^2 + b^2 > c^2 $$, the triangle is **acute**.
   - If $$ a^2 + b^2 = c^2 $$, the triangle is **right**.
   - If $$ a^2 + b^2 < c^2 $$, the triangle is **obtuse**.

4. **Conclusion:**
   - Here, $$ 481 > 289 $$, which means $$ a^2 + b^2 > c^2 $$.

**Therefore, the triangle with sides 15, 16, and 17 is an **acute** triangle.**

**Answer:**  
With sides 15, 16, and 17, the triangle is acute.
The triangle with sides 15, 16, and 17 is acute.

Determine if the following lengths make an acute, right or obtuse triangle. 7. 15, 16, 17.

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