Triangle Inequality Basics and Validity Checks

1) A triangle has sides 10, 5, and 5. Can this triangle exist?

Yes, it is isosceles.

No, because 10 ≥ 5 + 5.

Yes, it is equilateral.

Yes, it is scalene.

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

So, it doesn’t satisfy the triangle inequality.

Explanation

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

So, it doesn’t satisfy the triangle inequality.

2) If two sides of a triangle are 9 and 2, which inequality must the third side satisfy?

7 < x < 11

9 < x < 11

2 < x < 9

X < 7

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.

Explanation

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.

3) Which of the following is an impossible triangle?

7, 24, 25

8, 15, 20

10, 10, 19

3, 4, 9

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.

Explanation

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.

4) Which of the following sets of side lengths can form a triangle?

2, 3, 6

10, 3, 14

7, 2, 9

4, 5, 8

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.

Explanation

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.

5) If two sides of a triangle are 7 and 10, which of the following could be the third side?

3

12

17

18

The third side (x) must satisfy (10−7) < x < (10+7), or 3 < x < 17 .

Only 12 lies between 3 and 17.

Explanation

The third side (x) must satisfy (10−7) < x < (10+7), or 3 < x < 17 .

Only 12 lies between 3 and 17.

6) A triangle has side lengths 5 and 8. What is the greatest possible integer length for the third side?

12

13

14

15

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

The greatest integer less than 13 is 12.

Explanation

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

The greatest integer less than 13 is 12.

7) A triangle has sides 6, 6, and x. Which must be true?

X > 12

0 < x < 12

X > 6

X < 6

Sum of smaller sides > largest side → 6 + 6 > x, so x <12.

Also, x must be positive → 0 < x < 12 .

Explanation

Sum of smaller sides > largest side → 6 + 6 > x, so x <12.

Also, x must be positive → 0 < x < 12 .

8) Which of the following cannot form a triangle?

3, 4, 5

5, 5, 9

10, 6, 4

2, 7, 8

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.

Explanation

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.

9) If a triangle has sides 9 and 12, which of the following is not possible for the third side?

3

10

15

20

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

x = 3 does not satisfy the inequality.

Explanation

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

x = 3 does not satisfy the inequality.

10) What is the smallest integer value that could be the third side of a triangle with sides 11 and 15?

3

4

5

6

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.

Explanation

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.

11) The triangle inequality theorem states:

The sum of any two sides is greater than the third side.

The sum of any two sides is less than the third side.

The product of two sides is greater than the third side.

The difference of two sides equals the third side.

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.

Explanation

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.

12) Which of the following sets of numbers satisfies the triangle inequality?

6, 10, 3

12, 8, 5

15, 7, 8

20, 4, 15

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.

Explanation

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.

13) A triangle has sides 6, x, and 11. Which inequality is true?

5 < x < 17

6 < x < 11

0 < x < 6

17 < x < 20

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 .

Explanation

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 .

14) If exactly two sides of a triangle are equal, what type is it?

Equilateral

Isosceles

Scalene

Right

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.

Explanation

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.

15) Which set of numbers forms an equilateral triangle?

5, 5, 5

4, 4, 6

3, 3, 2

2, 4, 6

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.

Explanation

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.

16) Which of the following triples could be the sides of a scalene triangle?

5, 5, 5

4, 4, 7

6, 7, 10

9, 9, 12

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.

Explanation

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.

17) The sides of a triangle are 4, 4, and x. Which is true?

0 < x < 8

X > 8

X < 0

4 < x < 8

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

Hence, 0 < x < 8 .

Explanation

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

Hence, 0 < x < 8 .

18) If a triangle has sides 13 and 5, which of the following could be the third side?

4

9

18

20

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.

Explanation

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.

19) Which set of sides cannot form a triangle?

1, 1, 2

2, 2, 3

3, 4, 6

6, 8, 13

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.

Explanation

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.

20) If exactly two sides of a triangle are equal, what type is it?

Equilateral

Isosceles

Scalene

Right

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.

Explanation

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.