Triangle Inequality: Basics and Validity Checks

Check: 5 + 5 = 10, which equals 10, not greater.

So, it doesn’t satisfy the triangle inequality.

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, for sides of lengths 9 and 2, the third side (x) must satisfy the inequalities: x + 2 > 9 (which simplifies to x > 7) and x + 9 > 2 (which simplifies to x < 11). Thus, the valid range for x is 7 < x < 11.

An impossible triangle is a set of three sides where the sum of the lengths of any two sides is less than or equal to the length of the remaining side. In this case, option D (3, 4, 9) is impossible because 3 + 4 = 7, which is less than 9, making it impossible to form a triangle.

Use triangle inequality: sum of any two sides must be greater than the third. For option D (4, 5, 8): 4 + 5 > 8 (True), 4 + 8 > 5 (True), and 5 + 8 > 4 (True). Thus, a triangle can be formed.

The third side must be greater than the difference and less than the sum.

Step 1: 10 − 7 = 3 < x < 17 = 10 + 7.

Step 2: 12 fits in this range.

Result: x = 12 works.

The third side must be less than 5 + 8 = 13 and greater than 8 - 5 = 3.

The greatest integer less than 13 is 12.

The third side must be smaller than the sum, greater than the difference.

Step 1: 6 − 6 = 0 < x < 6 + 6 = 12.

To determine whether three lengths can form a triangle, we can use the triangle inequality theorem, which states that the sum of the lengths of any two sides must be greater than the length of the remaining side. For the option C (10, 6, 4): 10 is not less than (6 + 4), which equals 10. Therefore, these lengths cannot form a triangle.

The third side (x) must satisfy (12 − 9) < x < (12 + 9), or  3 < x < 21.

x = 3 does not satisfy the inequality.

The difference between sides is 15−11= 4 , and the sum is 26.

So, 4 < x < 26. The smallest integer greater than 4 is 5.

The sum of any two sides of a triangle must be greater than the third side.

That’s the fundamental rule that determines if a triangle can exist.

The triangle inequality states that for any triangle with sides of lengths a, b, and c, the following must hold: a + b > c, a + c > b, and b + c > a. In option B (12, 8, 5), the conditions are satisfied: 12 + 8 > 5, 12 + 5 > 8, and 8 + 5 > 12. Other options do not satisfy all three conditions.

In a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. For the triangle with sides 6, x, and 11, we apply the triangle inequality: 6 + x > 11 (which simplifies to x > 5) and 6 + 11 > x (which simplifies to x < 17). Therefore, the valid range for x is 5 < x < 17 .

A triangle with exactly two equal sides is known as an isosceles triangle. This type of triangle has two angles that are also equal, which is a defining property of isosceles triangles. In contrast, an equilateral triangle has all three sides equal, a scalene triangle has all sides of different lengths, and a right triangle has one angle that is 90 degrees.

An equilateral triangle is defined as a triangle in which all three sides are of equal length. In this case, the set of numbers 5, 5, 5 represents the lengths of the sides of the triangle, making it equilateral. The other options do not have equal lengths among their numbers, hence do not form an equilateral triangle.

A scalene triangle is defined as a triangle with all sides of different lengths. In this context, option C (6, 7, 10) has all distinct side lengths, which qualifies it as a scalene triangle. Options A, B, and D all contain at least two equal sides, making them isosceles triangles instead.

Two sides add up to 8, so x < 8 and x > 0 .

Hence, 0 < x < 8 .

In a triangle, the length of 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 must be less than 13 + 5 = 18 and greater than 13 - 5 = 8. Therefore, the possible values for the third side are between 8 and 18. Among the options, only 9 fits this criterion.

To determine if three lengths can form a triangle, we use 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. For the set (1, 1, 2), the sum of 1 and 1 is equal to 2, which does not satisfy the inequality (1 + 1 > 2). Therefore, this set cannot form a triangle.

A triangle with exactly two equal sides is known as an isosceles triangle. This type of triangle has two angles that are also equal, which is a defining property of isosceles triangles. In contrast, an equilateral triangle has all three sides equal, a scalene triangle has all sides of different lengths, and a right triangle has one angle that is 90 degrees.