Answer:
The triangle is acute.
Explanation:
To determine whether the side lengths of 3, 7, and 9 can form a triangle, we need to apply the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Let's check if this condition is satisfied for the given side lengths:
3 + 7 = 10 > 9, so the sum of the two smaller sides is greater than the largest side.
7 + 9 = 16 > 3, so the sum of the two smaller sides is greater than the largest side.
3 + 9 = 12 > 7, so the sum of the two smaller sides is greater than the largest side.
Since all three inequalities are true, we can conclude that the given side lengths of 3, 7, and 9 can form a triangle.
To determine whether the triangle is acute, right, or obtuse, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.
Let's first determine which side is the longest. Since 9 is the largest side length, it must be the hypotenuse if the triangle is right-angled.
However, using the Pythagorean theorem, we can see that:
3^2 + 7^2 = 58 < 9^2, so the triangle is not right-angled.
Therefore, the triangle must be either acute or obtuse. To determine which one, we can use the law of cosines, which states that in any triangle:
c^2 = a^2 + b^2 - 2ab*cos(C)
where a, b, and c are the side lengths, and C is the angle opposite to side c.
Let's apply this formula to the triangle with side lengths 3, 7, and 9:
9^2 = 3^2 + 7^2 - 2(3)(7)*cos(C)
81 = 58 - 42*cos(C)
cos(C) = (58 - 81)/(-42) = 0.5476
C = cos^(-1)(0.5476) = 56.69 degrees
Therefore, the largest angle in the triangle is approximately 56.69 degrees. Since this angle is less than 90 degrees, we can conclude that the triangle is acute.