Third Side Ranges and Edge Cases

Grade 7th

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| By Thames

Thames

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Quizzes Created: 7387 | Total Attempts: 9,527,684

| Questions: 20 | Updated: Nov 21, 2025

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1) A triangle has sides 8 and 5. Which of the following could be the third side?

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.

Explanation

Submit

About This 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?

5 < x < 13

4 < x < 9

9 < x < 13

0 < x < 5

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.

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.

Submit

3) Which of these sets can form a triangle?

2, 2, 4

6, 6, 12

5, 8, 13

4, 7, 10

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.

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.

Submit

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

5 < x < 25

10 < x < 15

0 < x < 5

X > 25

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.

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.

Submit

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

7, 10, 12

9, 9, 18

8, 15, 10

5, 6, 7

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.

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.

Submit

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

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.

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.

Submit

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

Again, the range is 4 < x < 18.

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

Explanation

Again, the range is 4 < x < 18.

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

Submit

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

6, 9, 15

9, 5, 13

12, 4, 20

8, 2, 6

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

All are true, so these sides make 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.

Submit

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

5 < x < 13

6 < x < 9

4 < x < 9

0 < x < 5

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

So the third side is between 5 and 13.

Explanation

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

So the third side is between 5 and 13.

Submit

10) Which of these is a valid triangle?

7, 24, 30

8, 10, 20

5, 6, 11

9, 2, 12

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

All three conditions are met.

Explanation

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

All three conditions are met.

Submit

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

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.

Explanation

Submit

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

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

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

Explanation

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

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

Submit

13) Which set of numbers makes an isosceles triangle?

6, 6, 9

7, 10, 12

5, 8, 12

9, 10, 11

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

Explanation

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

Submit

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

7 < x < 17

0 < x < 5

5 < x < 12

12 < x < 20

12 − 5 = 7 and 12 + 5 = 17.

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

Explanation

12 − 5 = 7 and 12 + 5 = 17.

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

Submit

15) Which set of sides forms a scalene triangle?

5, 5, 5

8, 8, 12

6, 7, 9

10, 10, 15

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.

Explanation

Submit

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

2, 3, 4

5, 6, 11

9, 9, 5

8, 9, 10

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.

Explanation

Submit

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

Right

Equilateral

Scalene

Isoceles

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.

Explanation

Submit

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

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.

Explanation

Submit

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

9, 12, 15

8, 8, 12

10, 6, 14

7, 10, 18

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.

Explanation

Submit

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

A + b > c

A + b < c

A = b + c

A × b > c

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

Explanation

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

Submit

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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...