Third Side Ranges and Edge Cases

  • 7th Grade
Reviewed by Cierra Henderson
Cierra Henderson, MBA |
K-12 Expert
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Cierra is an educational consultant and curriculum developer who has worked with students in K-12 for a variety of subjects including English and Math as well as test prep. She specializes in one-on-one support for students especially those with learning differences. She holds an MBA from the University of Massachusetts Amherst and a certificate in educational consulting from UC Irvine.
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| Attempts: 11 | Questions: 20 | Updated: Jan 16, 2026
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1) A triangle has sides 8 and 5. Which of the following could be the third side?

Explanation

In any triangle, one side must be less than the sum and greater than the difference of the other two sides.

Here, 8 + 5 = 13 and 8 − 5 = 3, so the third side must be greater than 3 and less than 13 (3 < x < 13).

Only 12 fits this range, so the third side could be 12 cm.

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About This Quiz
Third Side Ranges and EDGE Cases - Quiz

Get ready to discover how knowing just two sides of a triangle can tell you the range of the third! In this quiz, you’ll practice finding possible side lengths, ruling out impossible cases, and even classifying triangles along the way. Step by step, you’ll see how the triangle inequality helps... see moresolve tricky problems with missing sides.
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2) If two sides of a triangle are 9 and 4, what is the range for the third side?

Explanation

The difference is 9 − 4 = 5, and the sum is 9 + 4 = 13.

Therefore, the third side must be greater than 5 and less than 13.

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3) Which of these sets can form a triangle?

Explanation

For 4, 7, 10 → 4 + 7 = 11 > 10 ✅, 7 + 10 = 17 > 4 ✅, 4 + 10 = 14 > 7 ✅.

All are true, so this set forms a triangle.

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4) A triangle has sides 10, x, and 15. Which inequality is true?

Explanation

Subtract and add the known sides: 15 − 10 = 5 and 15 + 10 = 25.

So the third side must be greater than 5 and less than 25.

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5) Which set of side lengths cannot form a triangle?

Explanation

When we add the two shorter sides, 9 + 9 = 18.

Since the sum equals the third side (instead of being greater), the figure would be a straight line, not a triangle.

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6) If two sides of a triangle are 11 and 7, which is the largest possible integer length of the third side?

Explanation

11 + 7 = 18 and 11 − 7 = 4.

So the third side must be less than 18 and greater than 4.

The largest integer less than 18 is 17.

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7) If two sides of a triangle are 11 and 7, which is the smallest possible integer length of the third side?

Explanation

Again, the range is 4 < x < 18.

The smallest integer greater than 4 is 5, so the third side can be 5.

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8) Which of the following sets can form a triangle?

Explanation

Check sums: 9 + 5 = 14 > 13, 9 + 13 = 22 > 5, and 5 + 13 = 18 > 9.

All are true, so these sides make a triangle.

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9) A triangle has sides 4, x, and 9. Which inequality must hold?

Explanation

Subtract and add the known sides: 9 − 4 = 5 and 9 + 4 = 13.

So the third side is between 5 and 13.

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10) Which of these is a valid triangle?

Explanation

7 + 24 = 31 > 30 ✅, 24 + 30 = 54 > 7 ✅, and 7 + 30 = 37 > 24 ✅.

All three conditions are met.

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11) If a triangle has two sides of 6 and 14, which could be the third side?

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. In this case, 6 + 14 = 20, so the third side must be less than 20. Therefore, only 15 fits this condition.

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12) Which cannot be the third side of a triangle with sides 9 and 3?

Explanation

The range is 9 − 3 = 6 < x < 12 = 9 + 3.

15 is too large, so it’s not possible.

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13) Which set of numbers makes an isosceles triangle?

Explanation

A scalene triangle has all sides of different lengths & the set of sides 6, 7, and 9 are all distinct. 

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14) A triangle has sides 12, x, and 5. Which is true?

Explanation

12 − 5 = 7 and 12 + 5 = 17.

So the third side must be greater than 7 and less than 17.

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15) Which set of sides forms a scalene triangle?

Explanation

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

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16) Which of the following sets cannot form a triangle?

Explanation

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.

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17) If a triangle has only two equal sides, which of the following must it be classified as?

Explanation

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.

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18) If a triangle has sides 7 and 10, which is not possible for the third side?

Explanation

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.

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19) Which set of side lengths forms a right triangle?

Explanation

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, 9² + 12² = 81 + 144 = 225 = 15².

It fits the Pythagorean rule and the triangle inequality.

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20) Which inequality must always be true in any triangle with side lengths a, b, and c?

Explanation

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

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Cierra Henderson |MBA |
K-12 Expert
Cierra is an educational consultant and curriculum developer who has worked with students in K-12 for a variety of subjects including English and Math as well as test prep. She specializes in one-on-one support for students especially those with learning differences. She holds an MBA from the University of Massachusetts Amherst and a certificate in educational consulting from UC Irvine.
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A triangle has sides 8 and 5. Which of the following could be the...
If two sides of a triangle are 9 and 4, what is the range for the...
Which of these sets can form a triangle?
A triangle has sides 10, x, and 15. Which inequality is true?
Which set of side lengths cannot form a triangle?
If two sides of a triangle are 11 and 7, which is the largest possible...
If two sides of a triangle are 11 and 7, which is the smallest...
Which of the following sets can form a triangle?
A triangle has sides 4, x, and 9. Which inequality must hold?
Which of these is a valid triangle?
If a triangle has two sides of 6 and 14, which could be the third...
Which cannot be the third side of a triangle with sides 9 and 3?
Which set of numbers makes an isosceles triangle?
A triangle has sides 12, x, and 5. Which is true?
Which set of sides forms a scalene triangle?
Which of the following sets cannot form a triangle?
If a triangle has only two equal sides, which of the following must it...
If a triangle has sides 7 and 10, which is not possible for the third...
Which set of side lengths forms a right triangle?
Which inequality must always be true in any triangle with side lengths...
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