A triangle is classified based on its side lengths and angles. To determine the type of triangle formed by the side lengths of 4, 7, and 9, we can begin by checking whether these lengths satisfy the triangle inequality theorem. This theorem states that 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 the inequalities:
- 4 + 7 > 9 → 11 > 9 (True)
- 4 + 9 > 7 → 13 > 7 (True)
- 7 + 9 > 4 → 16 > 4 (True)
Since all three conditions are satisfied, a triangle can indeed be formed with these sides.
Next, we classify the triangle further:
- To determine whether it is a scalene, isosceles, or equilateral triangle, we look at the uniqueness of the side lengths:
- A triangle is equilateral if all three sides are equal. (Not the case here)
- A triangle is isosceles if it has at least two equal sides. (Not the case here)
- A triangle is scalene if all three sides are of different lengths. (This applies as 4, 7, and 9 are all different)
Therefore, the triangle formed by the side lengths of 4, 7, and 9 is a scalene triangle.
Moreover, we can also describe its angles. To find the type of angle (acute, obtuse, or right), we can utilize the Pythagorean theorem:
- If c is the longest side, the triangle is a right triangle if a2 + b2 = c2, acute if a2 + b2 > c2, and obtuse if a2 + b2 < c2.
In our case, let’s assign 9 as side c, then:
42 + 72 = 16 + 49 = 65
92 = 81
Since 65 < 81, this means we have an obtuse triangle.
In summary, the triangle with side lengths of 4, 7, and 9 is a scalene triangle with an obtuse angle.