Question

Classify each triangle as acute, right, or obtuse.
10, 16, 2sqrt39

Answer 1

To classify the triangle with side lengths \(10\), \(16\), and \(2\sqrt{39}\), we need to determine the relationship between the squares of the lengths of the sides.

1. First, let's denote the sides:
- \(a = 10\)
- \(b = 16\)
- \(c = 2\sqrt{39}\)

2. Calculate \(c^2\):
\[
c^2 = (2\sqrt{39})^2 = 4 \cdot 39 = 156
\]

3. Calculate \(a^2\) and \(b^2\):
\[
a^2 = 10^2 = 100
\]
\[
b^2 = 16^2 = 256
\]

4. Now, we compare \(a^2 + b^2\) with \(c^2\):
\[
a^2 + b^2 = 100 + 256 = 356
\]

5. Now we compare:
- If \(a^2 + b^2 < c^2\), the triangle is obtuse.
- If \(a^2 + b^2 = c^2\), the triangle is right.
- If \(a^2 + b^2 > c^2\), the triangle is acute.

Since \(356 > 156\), we conclude that:
\[
a^2 + b^2 > c^2
\]

Thus, the triangle is classified as **acute**.