8) Determine if the lengths 3,9 , and 10 form an acute, right, or obtuse triangle, or if they do-not form a triangle at all. not a triangle acute right obtuse

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To determine whether the lengths **3, 9, and 10** form an acute, right, or obtuse triangle (or do not form a triangle at all), follow these steps:

### 1. **Check the Triangle Inequality Theorem**
A set of three lengths can form a triangle if the sum of any two sides is greater than the third side.

- **3 + 9 > 10?**  
  $$ 12 > 10 $$ → **True**

- **3 + 10 > 9?**  
  $$ 13 > 9 $$ → **True**

- **9 + 10 > 3?**  
  $$ 19 > 3 $$ → **True**

Since all three conditions are true, the lengths **3, 9, and 10** can form a triangle.

### 2. **Determine the Type of Triangle**
To determine whether the triangle is acute, right, or obtuse, use the **Pythagorean Theorem**:

- Let **c** be the longest side. In this case, $$ c = 10 $$.
- Let **a** and **b** be the other two sides. Here, $$ a = 3 $$ and $$ b = 9 $$.

Compute $$ c^2 $$ and $$ a^2 + b^2 $$:

- $$ c^2 = 10^2 = 100 $$
- $$ a^2 + b^2 = 3^2 + 9^2 = 9 + 81 = 90 $$

**Compare the two values:**

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

Here, $$ 100 > 90 $$, so $$ c^2 > a^2 + b^2 $$.

### **Conclusion:**
The triangle with sides **3, 9, and 10** is an **obtuse** triangle.

**Answer:**  
**obtuse**
The triangle with sides 3, 9, and 10 is an obtuse triangle.

Determine if the lengths 3,9 , and 10 form an acute, right, or obtuse
triangle, or if they do-not form a triangle at all.
not a triangle
acute
right
obtuse

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