Question

Classify each triangle as acute, right or obtuse.
10,16, 2 square root of 39

the side lengths form a____ triangle

Answer 1

you sure

Answer 2

To classify the triangle with side lengths \( a = 10 \), \( b = 16 \), and \( c = 2\sqrt{39} \), we need to determine the type of triangle based on the angles created by these sides using the properties of triangle inequalities and the Pythagorean theorem.

First, let's find the length of \( 2\sqrt{39} \):

\[
2\sqrt{39} \approx 2 \times 6.244 \approx 12.488
\]

Now we have the triangle sides as approximately:
- \( a = 10 \)
- \( b = 16 \)
- \( c \approx 12.488 \)

To classify the triangle, we need to check if it is acute, right, or obtuse using the relationship between the squares of the side lengths:

1. For an **acute triangle**: \( a^2 + b^2 > c^2 \)
2. For a **right triangle**: \( a^2 + b^2 = c^2 \)
3. For an **obtuse triangle**: \( a^2 + b^2 < c^2 \)

Now, let's calculate \( a^2 \), \( b^2 \), and \( c^2 \):

\[
a^2 = 10^2 = 100
\]
\[
b^2 = 16^2 = 256
\]
\[
c^2 = (2\sqrt{39})^2 = 4 \times 39 = 156
\]

Next, let's check:

\[
a^2 + b^2 = 100 + 256 = 356
\]

Now, compare \( a^2 + b^2 \) to \( c^2 \):

\[
356 > 156
\]

Since \( a^2 + b^2 > c^2 \), the triangle is **acute**.

Thus, the side lengths form an **acute** triangle.

Answer 3

Let's double-check the calculations step by step to ensure the classification of the triangle with side lengths \( a = 10 \), \( b = 16 \), and \( c = 2\sqrt{39} \).

1. **Calculating \( c = 2\sqrt{39} \)**:

- \(\sqrt{39} \approx 6.244\)
- \(c = 2 \times 6.244 \approx 12.488\)

2. **Calculating the squares of the sides**:

- \(a^2 = 10^2 = 100\)
- \(b^2 = 16^2 = 256\)
- \(c^2 = (2\sqrt{39})^2 = 4 \times 39 = 156\)

3. **Calculating \( a^2 + b^2 \)**:

- \(a^2 + b^2 = 100 + 256 = 356\)

4. **Comparing \( a^2 + b^2 \) to \( c^2 \)**:

- \(356 > 156\)

Since \(a^2 + b^2 > c^2\), this means the triangle is **acute**.

### Conclusion:

Yes, the confirmation still holds that the triangle with side lengths 10, 16, and \( 2\sqrt{39} \) is an **acute triangle**. Thank you for your patience!