Question

The lengths of the sides of a triangle are StartRoot 39 EndRoot​, StartRoot 3 EndRoot​, and 6. Classify the triangle as open double quoteacuteclose double quote​, open double quoterightclose double quote​, or open double quoteobtuseclose double quote.
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Part 1
Choose the correct type of triangle below.
right
acute
obtuse

Answer 1

To classify the triangle with side lengths \(\sqrt{39}\), \(\sqrt{3}\), and \(6\), we first need to compare the squares of the lengths of the sides.

Let \(a = \sqrt{39}\), \(b = \sqrt{3}\), and \(c = 6\).

1. Calculate the squares of each side:
- \(a^2 = (\sqrt{39})^2 = 39\)
- \(b^2 = (\sqrt{3})^2 = 3\)
- \(c^2 = 6^2 = 36\)

2. To classify the triangle, we will use the relationship between the sides:
- 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.

Now, calculate \(a^2 + b^2\):
\[
a^2 + b^2 = 39 + 3 = 42
\]

Now we compare it with \(c^2\):
\[
42 > 36
\]

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

Thus, the correct type of triangle is **acute**.