To classify the triangle based on the given side lengths (12, 9, and 17), we can use the triangle inequality theorem and the properties of triangles with respect to their angles.
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Triangle Inequality Theorem: For any triangle with sides \( a \), \( b \), and \( c \) (where \( c \) is the longest side), the following must hold:
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)
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Check the Triangle Inequality:
- Let \( a = 9 \), \( b = 12 \), and \( c = 17 \) (as 17 is the longest side).
- Check \( a + b > c \): \( 9 + 12 = 21 > 17 \) (this holds)
- Check \( a + c > b \): \( 9 + 17 = 26 > 12 \) (this holds)
- Check \( b + c > a \): \( 12 + 17 = 29 > 9 \) (this holds)
Since all conditions of the triangle inequality are satisfied, the sides can indeed form a triangle.
- Type of Triangle: To classify the type of triangle based on its angles, we can use the relationship involving the squares of 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.
Calculate:
- \( a^2 = 9^2 = 81 \)
- \( b^2 = 12^2 = 144 \)
- \( c^2 = 17^2 = 289 \)
Now check:
- \( a^2 + b^2 = 81 + 144 = 225 \)
- Compare with \( c^2 \): \( 225 < 289 \)
Since \( a^2 + b^2 < c^2 \), the triangle is classified as obtuse.
Thus, the side lengths \( 12, 9, 17 \) form an obtuse triangle.