To classify the triangle with side lengths 12, 9, and 17, we can use the Pythagorean theorem.
First, let’s check if these lengths satisfy the triangle inequality, which states that the sum of the lengths of any two sides must be greater than the length of the third side:
- \( 12 + 9 > 17 \) (21 > 17) - true
- \( 12 + 17 > 9 \) (29 > 9) - true
- \( 9 + 17 > 12 \) (26 > 12) - true
Since all inequalities are satisfied, a triangle can be formed.
Next, we determine whether it is acute, right, or obtuse by comparing the squares of the lengths. In a triangle with sides \( a \), \( b \), and \( c \) (where \( c \) is the longest side), we check the following:
- If \( a^2 + b^2 > c^2 \), it is an acute triangle.
- If \( a^2 + b^2 = c^2 \), it is a right triangle.
- If \( a^2 + b^2 < c^2 \), it is an obtuse triangle.
Here, we have:
- \( a = 9 \)
- \( b = 12 \)
- \( c = 17 \)
Now let's calculate:
- \( a^2 + b^2 = 9^2 + 12^2 = 81 + 144 = 225 \)
- \( c^2 = 17^2 = 289 \)
Now compare:
- \( 225 < 289 \)
Since \( a^2 + b^2 < c^2 \), the triangle is obtuse.
Therefore, these side lengths form an obtuse triangle.
1 answer