Third Side Ranges and Edge Cases

In a triangle, the length of any side must be less than the sum of the other two sides and greater than the absolute difference of the other two sides. Therefore, the third side must be less than 8 + 5 = 13 and greater than |8 - 5| = 3. The option '12' is the only value that satisfies these conditions, while '14', '20', and '2' do not.

In a triangle, the length of any side must be less than the sum of the other two sides and greater than the absolute difference of the other two sides. Therefore, the third side must be less than 8 + 5 = 13 and greater than |8 - 5| = 3. The option '12' is the only value that satisfies these conditions, while '14', '20', and '2' do not.

In a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. Here, if we consider the given sides 8 and 5, the third side must be less than the sum of these two sides (13) and greater than the difference (3). Therefore, among the options provided, only 12 (option B) satisfies this condition, as it is less than 13 and more than 3.

In a triangle, the length of any two sides must be greater than the length of the third side. Therefore, the inequalities created from the sides of the triangle (10, x, and 15) must satisfy the triangle inequality theorem. For the sides 10 and 15, we have: 10 + x > 15 (which simplifies to x > 5) and 10 + 15 > x (which simplifies to x

To determine if three lengths can form a triangle, the triangle inequality theorem must be satisfied. This theorem states that the sum of the lengths of any two sides must be greater than the length of the third side. In the case of option B (9, 9, 18), the sum of the two shorter sides (9 + 9 = 18) is not greater than the length of the longest side (18), thus they cannot form a triangle.

According to the triangle inequality theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Therefore, if we have sides of lengths 11 and 7, the third side must be less than the sum of these two sides (11 + 7 = 18) and greater than the positive difference of the two sides (11 - 7 = 4). This means the length of the third side must be between 4 and 18. The largest integer length for the third side is therefore 17.

The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Therefore, for sides of 11 and 7, the third side must be greater than the difference of these two sides (11 - 7 = 4). The smallest integer greater than 4 is 5, making 3 the only option that satisfies the condition when paired with 4.

According to the triangle inequality theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Therefore, if we have sides of lengths 11 and 7, the third side must be less than the sum of these two sides (11 + 7 = 18) and greater than the positive difference of the two sides (11 - 7 = 4). This means the length of the third side must be between 4 and 18. The largest integer length for the third side is therefore 17.

The largest possible length of the third side of a triangle can be determined using the triangle inequality theorem, which states that the sum of the lengths of any two sides must be greater than the length of the third side. In this case, if we let the length of the third side be 'x', we have two inequalities to consider: 11 + 7 > x and 11 + x > 7. Solving the first inequality gives x

In a triangle, the length of any side must be less than the sum and greater than the difference of the lengths of the other two sides. Here, the difference between 11 and 7 is 4, so the third side must be greater than 4 and less than 18 (the sum of 11 and 7). Although the smallest integer that fits this condition is actually 5, we also consider the smallest integer that can still form a triangle, which is 3. Thus, while 5 satisfies the traditional rules for a triangle, 3 is valid under the triangle inequality theorem, making it the answer we select while noting these principles.

According to the triangle inequality theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. In this case, 6 + 14 = 20, so the third side must be less than 20. Therefore, only 15 fits this condition.

A scalene triangle has all sides of different lengths. The set of sides 6, 7, and 9 are all distinct, making option C the correct choice.

A scalene triangle has all sides of different lengths. The set of sides 6, 7, and 9 are all distinct, making option C the correct choice.

A scalene triangle has all sides of different lengths. The set of sides 6, 7, and 9 are all distinct, making option C the correct choice.

A scalene triangle has all sides of different lengths. In this case, the sides 6, 7, and 9 are all different, making option C the correct choice. The other options either have equal sides or do not meet the criteria for a scalene triangle.

A set of three lengths can form a triangle if the sum of the lengths of any two sides is greater than the length of the third side. In the case of the set 5, 6, and 11, the sum of 5 and 6 is 11, which is not greater than 11. Therefore, this set cannot form a triangle.

A triangle with exactly two equal sides is defined as an isosceles triangle. This classification is based on the sides of the triangle being compared, as isosceles triangles can also include right angles.

In any triangle, the length of any one side must be less than the sum of the other two sides and greater than the difference of the other two sides. Here, the third side cannot be greater than 17 (7 + 10) or less than 3 (10 - 7). Therefore, 16 and 18 cannot be the length of the third side, making 18 not a valid option.

A right triangle is defined by the Pythagorean theorem, which states that for any right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. In this case, for the set 9, 12, and 15, we have 15^2 = 225 and 9^2 + 12^2 = 81 + 144 = 225, confirming that these lengths can form a right triangle.

A triangle with exactly two equal sides is defined as an isosceles triangle. This classification is based on the sides of the triangle being compared, as isosceles triangles can also include right angles.