To classify the triangle with side lengths 9.6, 18, and 20.1, we can use the triangle inequality theorem and the properties of triangle angles.
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Triangle Inequality Theorem:
- For any triangle, the sum of the lengths of any two sides must be greater than the length of the third side.
Let's check this:
- \(9.6 + 18 = 27.6 > 20.1\) (true)
- \(9.6 + 20.1 = 29.7 > 18\) (true)
- \(18 + 20.1 = 38.1 > 9.6\) (true)
The sides satisfy the triangle inequality, so they can form a triangle.
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Classifying the triangle based on angles:
- We can use the Pythagorean theorem, which states that in a right triangle, the square of the longest side (hypotenuse) is equal to the sum of the squares of the other two sides. If the square of the longest side is less than the sum of the squares of the other two sides, the triangle is acute. If it is greater, the triangle is obtuse.
The longest side is 20.1. We will check the following:
\[ a = 9.6, \quad b = 18, \quad c = 20.1 \]
We check:
- \(c^2\) vs \(a^2 + b^2\):
- \(20.1^2 = 404.01\)
- \(9.6^2 + 18^2 = 92.16 + 324 = 416.16\)
Now we compare:
- \(404.01 < 416.16\)
Since \(c^2 < a^2 + b^2\), the triangle is classified as acute.
Therefore, the triangle with side lengths 9.6, 18, and 20.1 forms an acute triangle.