Determine if the following lengths make an acute, right or obtuse triangle. 8. $$ 15,15,15 \sqrt{2} $$.

Question

UpStudy AI · Accepted Answer

To determine whether the triangle with side lengths $$15$$, $$15$$, and $$15\sqrt{2}$$ is acute, right, or obtuse, we can use the **Law of Cosines**.

### Step 1: Identify the Longest Side
In this case, the longest side is $$15\sqrt{2}$$.

### Step 2: Apply the Law of Cosines
For a triangle with sides $$a$$, $$b$$, and $$c$$ (where $$c$$ is the longest side):

- **Acute Triangle**: $$c^2 < a^2 + b^2$$
- **Right Triangle**: $$c^2 = a^2 + b^2$$
- **Obtuse Triangle**: $$c^2 > a^2 + b^2$$

Let's calculate:

1. $$c^2 = (15\sqrt{2})^2 = 225 	imes 2 = 450$$
2. $$a^2 + b^2 = 15^2 + 15^2 = 225 + 225 = 450$$

### Step 3: Compare the Values
Since $$c^2 = a^2 + b^2$$ (i.e., $$450 = 450$$), the triangle satisfies the condition for a **right triangle**.

### Conclusion
The triangle with sides $$15$$, $$15$$, and $$15\sqrt{2}$$ is a **right triangle**.

**Answer:** It is a right triangle.
The triangle is a right triangle.

Determine if the following lengths make an acute, right or obtuse triangle. 8. \( 15,15,15 \sqrt{2} \).