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.
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First, let's denote the sides:
- \(a = 10\)
- \(b = 16\)
- \(c = 2\sqrt{39}\)
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Calculate \(c^2\): \[ c^2 = (2\sqrt{39})^2 = 4 \cdot 39 = 156 \]
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Calculate \(a^2\) and \(b^2\): \[ a^2 = 10^2 = 100 \] \[ b^2 = 16^2 = 256 \]
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Now, we compare \(a^2 + b^2\) with \(c^2\): \[ a^2 + b^2 = 100 + 256 = 356 \]
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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.