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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

1 answer

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.