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Math › Geometry › Triangles › Triangle Inequality

Triangle Inequality in Real Situations

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| By Thames

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| Questions: 20

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1) Maria wants to build a triangular garden with side lengths 8 ft, 12 ft, and 5 ft. Will these dimensions work?

Yes, because 8 + 5 > 12

No, because 8 + 5 = 12

Yes, because 12 + 5 < 8

No, because 8 + 12 < 5

In order for three side lengths to form a triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. Here, 8 + 5 = 13, which is greater than 12, thus the dimensions can indeed form a triangle.

Explanation

In order for three side lengths to form a triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. Here, 8 + 5 = 13, which is greater than 12, thus the dimensions can indeed form a triangle.

Submit

About This Quiz

Triangle Inequality In Real Situations - Quiz

Are you ready to see how the triangle inequality connects to real-life situations? In this quiz, you’ll solve problems about gardens, ropes, beams, and even parks to decide if certain side lengths can form a triangle. You’ll practice checking the rule with real examples, spot impossible triangles, and classify special... see moreones like isosceles and right. Step by step, you’ll discover how this simple geometry rule explains problems you might run into every day!
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2) A triangle has sides 9 cm, 14 cm, and x cm. Which inequality describes x?

5 < x < 23

0 < x < 5

9 < x < 14

X > 23

In a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. Thus, for sides of lengths 9 cm and 14 cm, the inequalities are 9 + 14 > x (which simplifies to x 14 (which simplifies to x > 5). Therefore, the combined inequalities give us 5

Explanation

In a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. Thus, for sides of lengths 9 cm and 14 cm, the inequalities are 9 + 14 > x (which simplifies to x 14 (which simplifies to x > 5). Therefore, the combined inequalities give us 5

Submit

3) A builder uses beams of lengths 6 m and 8 m for two sides of a triangular frame. Which length could be the third beam?

2 m

3 m

15 m

17 m

To form a triangle, the length of the third side must be less than the sum of the other two sides (6 m + 8 m = 14 m) and greater than the absolute difference of the other two sides (|6 m - 8 m| = 2 m). Therefore, the possible lengths for the third beam must be greater than 2 m and less than 14 m. Among the options, only 3 m fits this criterion.

Explanation

To form a triangle, the length of the third side must be less than the sum of the other two sides (6 m + 8 m = 14 m) and greater than the absolute difference of the other two sides (|6 m - 8 m| = 2 m). Therefore, the possible lengths for the third beam must be greater than 2 m and less than 14 m. Among the options, only 3 m fits this criterion.

Submit

4) A triangle has sides 9 cm, 14 cm, and x cm. Which inequality describes x?

5 < x < 23

0 < x < 5

9 < x < 14

X > 23

In a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. Thus, for sides of lengths 9 cm and 14 cm, the inequalities are 9 + 14 > x (which simplifies to x 14 (which simplifies to x > 5). Therefore, the combined inequalities give us 5

Explanation

In a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. Thus, for sides of lengths 9 cm and 14 cm, the inequalities are 9 + 14 > x (which simplifies to x 14 (which simplifies to x > 5). Therefore, the combined inequalities give us 5

Submit

5) Which of the following sets of side lengths forms a scalene triangle?

, 5, 5

7, 7, 10

6, 7, 8

8, 8, 12

A scalene triangle is defined as a triangle with all sides of different lengths. In this case, the side lengths 6, 7, and 8 are all distinct, making option C the correct choice. Options A, B, and D all contain at least two sides of equal length, which do not satisfy the scalene triangle condition.

Explanation

A scalene triangle is defined as a triangle with all sides of different lengths. In this case, the side lengths 6, 7, and 8 are all distinct, making option C the correct choice. Options A, B, and D all contain at least two sides of equal length, which do not satisfy the scalene triangle condition.

Submit

6) Maria wants to build a triangular garden with side lengths 8 ft, 12 ft, and 5 ft. Will these dimensions work?

Yes, because 8 + 5 > 12

No, because 8 + 5 = 12

Yes, because 12 + 5 < 8

No, because 8 + 12 < 5

In order for three side lengths to form a triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. Here, 8 + 5 = 13, which is greater than 12, thus the dimensions can indeed form a triangle.

Explanation

In order for three side lengths to form a triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. Here, 8 + 5 = 13, which is greater than 12, thus the dimensions can indeed form a triangle.

Submit

7) A rope is cut into three pieces measuring 4 m, 9 m, and 13 m. Can these form a triangle?

Yes, because all sums are greater than the third side:

No, because 4 + 9 = 13:

No, because 9 + 13 < 4:

Yes, because 4 + 13 > 9:

To form a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. In this case, 4 m + 9 m equals 13 m, which is not greater, hence they cannot form a triangle.

Explanation

To form a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. In this case, 4 m + 9 m equals 13 m, which is not greater, hence they cannot form a triangle.

Submit

8) A triangle has two sides measuring 7 cm and 10 cm. Which of the following is a possible third side?

2 cm

18 cm

12 cm

20 cm

For a triangle, the length of any side must be less than the sum and greater than the difference of the other two sides. Here, the sum of 7 cm and 10 cm is 17 cm, and the difference is 3 cm. Therefore, the third side must be greater than 3 cm and less than 17 cm. Among the options, only 12 cm fits this criterion.

Explanation

For a triangle, the length of any side must be less than the sum and greater than the difference of the other two sides. Here, the sum of 7 cm and 10 cm is 17 cm, and the difference is 3 cm. Therefore, the third side must be greater than 3 cm and less than 17 cm. Among the options, only 12 cm fits this criterion.

Submit

9) John states 9, 12, and 20 can form a triangle. Is he correct?

Yes, because all conditions are met:

No, because 9 + 12 C 20:

Yes, because 9 + 20 > 12:

No, because 12 + 20 < 9:

For three lengths to form a triangle, the sum of any two lengths must be greater than the third. In this case, 9 + 12 is equal to 21, which is greater than 20, making it possible to form a triangle.

Explanation

For three lengths to form a triangle, the sum of any two lengths must be greater than the third. In this case, 9 + 12 is equal to 21, which is greater than 20, making it possible to form a triangle.

Submit

10) A builder uses beams of lengths 6 m and 8 m for two sides of a triangular frame. Which length could be the third beam?

2 m

3 m

15 m

17 m

To form a triangle, the length of the third side must be less than the sum of the other two sides (6 m + 8 m = 14 m) and greater than the absolute difference of the other two sides (|6 m - 8 m| = 2 m). Therefore, the possible lengths for the third beam must be greater than 2 m and less than 14 m. Among the options, only 3 m fits this criterion.

Explanation

To form a triangle, the length of the third side must be less than the sum of the other two sides (6 m + 8 m = 14 m) and greater than the absolute difference of the other two sides (|6 m - 8 m| = 2 m). Therefore, the possible lengths for the third beam must be greater than 2 m and less than 14 m. Among the options, only 3 m fits this criterion.

Submit

11) A triangle has sides 9 cm, 14 cm, and x cm. Which inequality describes x?

5 < x < 23

0 < x < 5

9 < x < 14

X > 23

In a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. Thus, for sides of lengths 9 cm and 14 cm, the inequalities are 9 + 14 > x (which simplifies to x 14 (which simplifies to x > 5). Therefore, the combined inequalities give us 5

Explanation

In a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. Thus, for sides of lengths 9 cm and 14 cm, the inequalities are 9 + 14 > x (which simplifies to x 14 (which simplifies to x > 5). Therefore, the combined inequalities give us 5

Submit

12) Can a triangle have sides of 10 in, 5 in, and 16 in?

Yes, because 10 + 5 > 16

No, because 10 + 5 = 16

No, because 10 + 5 < 16

Yes, because 16 < 10 + 5

A triangle cannot have sides that do not satisfy 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, 10 + 5 equals 15, which is less than 16, therefore it is not possible to form a triangle with these side lengths.

Explanation

A triangle cannot have sides that do not satisfy 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, 10 + 5 equals 15, which is less than 16, therefore it is not possible to form a triangle with these side lengths.

Submit

13) A triangle has sides 9 cm, 14 cm, and x cm. Which inequality describes x?

5 < x < 23

0 < x < 5

9 < x < 14

X > 23

In a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. Thus, for sides 9 cm and 14 cm, we have the inequalities 9 + 14 > x (x 14 (x > 5). Therefore, combining these gives the inequality 5

Explanation

In a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. Thus, for sides 9 cm and 14 cm, we have the inequalities 9 + 14 > x (x 14 (x > 5). Therefore, combining these gives the inequality 5

Submit

14) A triangle has sides 9 cm, 14 cm, and x cm. Which inequality describes x?

5 < x < 23

0 < x < 5

9 < x < 14

X > 23

In a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. Thus, for sides of lengths 9 cm and 14 cm, the inequalities are 9 + 14 > x (which simplifies to x 14 (which simplifies to x > 5). Therefore, the combined inequalities give us 5

Explanation

In a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. Thus, for sides of lengths 9 cm and 14 cm, the inequalities are 9 + 14 > x (which simplifies to x 14 (which simplifies to x > 5). Therefore, the combined inequalities give us 5

Submit

15) If a triangle has sides 4 cm, 7 cm, and 10 cm, which statement is true?

It is not a triangle because 4 + 7 < 10:

It is not a triangle because 4 + 10 = 7:

It is not a triangle because 7 + 10 < 4:

It is a triangle because 4 + 7 > 10:

A triangle can exist if the sum of any two sides is greater than the third side. Here, 4 cm + 7 cm is greater than 10 cm, confirming that it can form a triangle.

Explanation

A triangle can exist if the sum of any two sides is greater than the third side. Here, 4 cm + 7 cm is greater than 10 cm, confirming that it can form a triangle.

Submit

16) A triangle has sides 9 cm, 14 cm, and x cm. Which inequality describes x?

5 < x < 23

0 < x < 5

9 < x < 14

X > 23

In a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. Thus, for sides of lengths 9 cm and 14 cm, the inequalities are 9 + 14 > x (which simplifies to x 14 (which simplifies to x > 5). Therefore, the combined inequalities give us 5

Explanation

In a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. Thus, for sides of lengths 9 cm and 14 cm, the inequalities are 9 + 14 > x (which simplifies to x 14 (which simplifies to x > 5). Therefore, the combined inequalities give us 5

Submit

17) A triangle has sides 9 cm, 14 cm, and x cm. Which inequality describes x?

5 < x < 23

0 < x < 5

9 < x < 14

X > 23

In a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. Thus, for sides of lengths 9 cm and 14 cm, the inequalities are 9 + 14 > x (which simplifies to x 14 (which simplifies to x > 5). Therefore, the combined inequalities give us 5

Explanation

In a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. Thus, for sides of lengths 9 cm and 14 cm, the inequalities are 9 + 14 > x (which simplifies to x 14 (which simplifies to x > 5). Therefore, the combined inequalities give us 5

Submit

18) A triangle has two sides measuring 7 cm and 10 cm. Which of the following is a possible third side?

2 cm

18 cm

12 cm

20 cm

In a triangle, the length of the third side must be less than the sum of the other two sides and greater than the difference of the two sides. Here, 7 cm + 10 cm = 17 cm and |10 cm - 7 cm| = 3 cm. Therefore, the third side must be between 3 cm and 17 cm, making 12 cm a valid option.

Explanation

In a triangle, the length of the third side must be less than the sum of the other two sides and greater than the difference of the two sides. Here, 7 cm + 10 cm = 17 cm and |10 cm - 7 cm| = 3 cm. Therefore, the third side must be between 3 cm and 17 cm, making 12 cm a valid option.

Submit

19) A triangle has sides 9 cm, 14 cm, and x cm. Which inequality describes x?

5 < x < 23

0 < x < 5

9 < x < 14

X > 23

In a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. Thus, for sides 9 cm and 14 cm, we have the inequalities 9 + 14 > x (x 14 (x > 5). Therefore, combining these gives the inequality 5

Explanation

In a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. Thus, for sides 9 cm and 14 cm, we have the inequalities 9 + 14 > x (x 14 (x > 5). Therefore, combining these gives the inequality 5

Submit

20) If a triangle has sides 4 cm, 7 cm, and 10 cm, which statement is true?

It is not a triangle because 4 + 7 < 10:

It is not a triangle because 4 + 10 = 7:

It is not a triangle because 7 + 10 < 4:

It is a triangle because 4 + 7 > 10:

A triangle can exist if the sum of any two sides is greater than the third side. Here, 4 cm + 7 cm is greater than 10 cm, confirming that it can form a triangle.

Explanation

A triangle can exist if the sum of any two sides is greater than the third side. Here, 4 cm + 7 cm is greater than 10 cm, confirming that it can form a triangle.

Submit

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Oct 11, 2025

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