Triangle Inequality Quiz: Master Triangle Inequality Quiz

1) For sides 6, 8, and x to form a triangle, which must be true?

1 less than x less than 14

2 less than x less than 14

2 less than or equal to x less than 14

2 less than x less than or equal to 14

Lower bound: the absolute value of 8 minus 6 equals 2. Upper bound: 8+6=14. Both bounds are strict because equality produces a degenerate case. Therefore 2 less than x less than 14. Option A uses 1 as the lower bound, allowing invalid values. Options C and D use non-strict inequalities at one bound, incorrectly allowing x=2 or x=14.

Explanation

Lower bound: the absolute value of 8 minus 6 equals 2. Upper bound: 8+6=14. Both bounds are strict because equality produces a degenerate case. Therefore 2 less than x less than 14. Option A uses 1 as the lower bound, allowing invalid values. Options C and D use non-strict inequalities at one bound, incorrectly allowing x=2 or x=14.

2) Which set of side lengths forms a valid triangle?

3, 3, 2008

4, 5, 2009

6, 10, 17

7, 10, 16

Option D: 7+10=17>16, 7+16=23>10, 10+16=26>7, all inequalities hold — valid triangle. Option A: 3+3=6 which is much less than 2008 — invalid. Option B: 4+5=9 which is much less than 2009 — invalid. Option C: 6+10=16 which is not greater than 17, failing the triangle inequality. Only option D satisfies all three conditions.

Explanation

Option D: 7+10=17>16, 7+16=23>10, 10+16=26>7, all inequalities hold — valid triangle. Option A: 3+3=6 which is much less than 2008 — invalid. Option B: 4+5=9 which is much less than 2009 — invalid. Option C: 6+10=16 which is not greater than 17, failing the triangle inequality. Only option D satisfies all three conditions.

3) A triangle has sides 5, 10, and x. Which must be true?

5 less than x less than 15

0 less than x less than 5

10 less than x less than 15

X greater than 15

Lower bound: the absolute value of 10 minus 5 equals 5. Upper bound: 10+5=15. Both bounds are strict, giving 5 less than x less than 15. Option B restricts x below 5, all of which are invalid. Option C only covers the upper portion of the valid range, missing valid values between 5 and 10. Option D requires x to exceed 15, violating the upper bound.

Explanation

Lower bound: the absolute value of 10 minus 5 equals 5. Upper bound: 10+5=15. Both bounds are strict, giving 5 less than x less than 15. Option B restricts x below 5, all of which are invalid. Option C only covers the upper portion of the valid range, missing valid values between 5 and 10. Option D requires x to exceed 15, violating the upper bound.

4) A triangle has sides 9, 14, and x. Which inequality must hold?

5 less than x less than 23

0 less than x less than 9

12 less than x less than 21

X greater than 23

Lower bound: the absolute value of 14 minus 9 equals 5. Upper bound: 14+9=23. Therefore 5 less than x less than 23. Option B restricts x below 9, missing valid values above 9. Option C covers only a portion of the valid range. Option D requires x to exceed 23, violating the triangle inequality since x must be strictly less than 23.

Explanation

Lower bound: the absolute value of 14 minus 9 equals 5. Upper bound: 14+9=23. Therefore 5 less than x less than 23. Option B restricts x below 9, missing valid values above 9. Option C covers only a portion of the valid range. Option D requires x to exceed 23, violating the triangle inequality since x must be strictly less than 23.

5) In any triangle, the longest side is less than the sum of the other two sides.

True

False

The answer is True. Let c be the longest side. The triangle inequality requires a+b greater than c for all three combinations. Since c is the longest side, the condition a+b greater than c is the binding constraint. If it holds, the other two inequalities automatically hold as well. This is why the triangle inequality is often stated as: the longest side must be less than the sum of the other two.

Explanation

The answer is True. Let c be the longest side. The triangle inequality requires a+b greater than c for all three combinations. Since c is the longest side, the condition a+b greater than c is the binding constraint. If it holds, the other two inequalities automatically hold as well. This is why the triangle inequality is often stated as: the longest side must be less than the sum of the other two.

6) A triangle has two equal sides of 6 and 6. Which could be the third side?

12

11

0

13

Range: the absolute value of 6 minus 6 is less than x which is less than 6 plus 6, giving 0 less than x less than 12. Only 11 lies strictly between 0 and 12. Option A gives 12, which equals the upper bound and produces a degenerate case. Option C gives 0, which is the lower bound and also invalid. Option D gives 13, exceeding the upper bound.

Explanation

Range: the absolute value of 6 minus 6 is less than x which is less than 6 plus 6, giving 0 less than x less than 12. Only 11 lies strictly between 0 and 12. Option A gives 12, which equals the upper bound and produces a degenerate case. Option C gives 0, which is the lower bound and also invalid. Option D gives 13, exceeding the upper bound.

7) Select all sets that CAN form a triangle.

2, 2, 2005

7, 24, 25

9, 11, 15

3, 4, 2007

Option B: 7+24=31>25, 7+25=32>24, 24+25=49>7, all hold — valid triangle. Option C: 9+11=20>15, 9+15=24>11, 11+15=26>9, all hold — valid triangle. Option A: 2+2=4 which is much less than 2005 — cannot form a triangle. Option D: 3+4=7, which equals the third side rather than strictly exceeding it — cannot form a triangle.

Explanation

Option B: 7+24=31>25, 7+25=32>24, 24+25=49>7, all hold — valid triangle. Option C: 9+11=20>15, 9+15=24>11, 11+15=26>9, all hold — valid triangle. Option A: 2+2=4 which is much less than 2005 — cannot form a triangle. Option D: 3+4=7, which equals the third side rather than strictly exceeding it — cannot form a triangle.

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8) With sides 11 and 7, what is the smallest possible integer length of the third side?

4

5

6

7

Lower bound: the absolute value of 11 minus 7 equals 4, but x must be strictly greater than 4. The smallest integer satisfying x greater than 4 is 5. Option A gives 4, which equals the lower bound and is not valid since the inequality is strict. Options C and D give values that satisfy the inequality but are not the smallest possible integer.

Explanation

Lower bound: the absolute value of 11 minus 7 equals 4, but x must be strictly greater than 4. The smallest integer satisfying x greater than 4 is 5. Option A gives 4, which equals the lower bound and is not valid since the inequality is strict. Options C and D give values that satisfy the inequality but are not the smallest possible integer.

9) With sides 11 and 7, what is the largest possible integer length of the third side?

15

16

17

18

Upper bound: 11+7=18, but x must be strictly less than 18. The largest integer satisfying x less than 18 is 17. Option D gives 18, which equals the upper bound and produces a degenerate case. Options A and B give smaller integers that are valid but not the largest possible.

Explanation

Upper bound: 11+7=18, but x must be strictly less than 18. The largest integer satisfying x less than 18 is 17. Option D gives 18, which equals the upper bound and produces a degenerate case. Options A and B give smaller integers that are valid but not the largest possible.

10) Which set could form a right triangle?

6, 8, 10

7, 8, 12

5, 5, 12

4, 5, 10

Check option A: 6 squared plus 8 squared equals 36 plus 64 equals 100 equals 10 squared, satisfying the Pythagorean theorem. Option B: 7 squared plus 8 squared equals 113 which does not equal 144. Option C: 5+5=10 which is less than 12, failing the triangle inequality entirely. Option D: 4+5=9 which is less than 10, also failing.

Explanation

Check option A: 6 squared plus 8 squared equals 36 plus 64 equals 100 equals 10 squared, satisfying the Pythagorean theorem. Option B: 7 squared plus 8 squared equals 113 which does not equal 144. Option C: 5+5=10 which is less than 12, failing the triangle inequality entirely. Option D: 4+5=9 which is less than 10, also failing.

11) Which set of side lengths can form a triangle?

2, 3, 6

4, 7, 12

5, 9, 13

8, 10, 25

Check option C: 5+9=14>13, 5+13=18>9, 9+13=22>5. All three inequalities hold so 5, 9, 13 forms a valid triangle. Option A: 2+3=5 which is not greater than 6, failing the test. Option B: 4+7=11 which is less than 12, failing. Option D: 8+10=18 which is less than 25, failing.

Explanation

Check option C: 5+9=14>13, 5+13=18>9, 9+13=22>5. All three inequalities hold so 5, 9, 13 forms a valid triangle. Option A: 2+3=5 which is not greater than 6, failing the test. Option B: 4+7=11 which is less than 12, failing. Option D: 8+10=18 which is less than 25, failing.

12) If one side equals the sum of the other two, a valid triangle exists.

True

False

The answer is False. The triangle inequality requires each side to be strictly less than the sum of the other two. If a equals b plus c exactly, the three vertices would be collinear and the shape degenerates into a straight line segment with no enclosed area. The inequality must be strict: a less than b plus c, not a less than or equal to b plus c.

Explanation

The answer is False. The triangle inequality requires each side to be strictly less than the sum of the other two. If a equals b plus c exactly, the three vertices would be collinear and the shape degenerates into a straight line segment with no enclosed area. The inequality must be strict: a less than b plus c, not a less than or equal to b plus c.

13) If two sides are 10 and 4, which third side works?

4

5

6

12

Range: the absolute value of 10 minus 4 is less than x which is less than 10 plus 4, giving 6 less than x less than 14. Only 12 lies strictly between 6 and 14. Option A gives 4, which is less than 6. Option B gives 5, also less than 6. Option C gives 6, which equals the lower bound and is not valid since the inequality is strict.

Explanation

Range: the absolute value of 10 minus 4 is less than x which is less than 10 plus 4, giving 6 less than x less than 14. Only 12 lies strictly between 6 and 14. Option A gives 4, which is less than 6. Option B gives 5, also less than 6. Option C gives 6, which equals the lower bound and is not valid since the inequality is strict.

14) Select all sets that CANNOT form a triangle.

5, 6, 12

9, 10, 15

7, 7, 14

8, 8, 16

Option A: 5+6=11 which is less than 12, failing the inequality. Option C: 7+7=14, which equals the third side rather than strictly exceeding it. Option D: 8+8=16, also equal rather than strictly greater. All three cannot form a triangle. Option B: 9+10=19>15, 9+15=24>10, 10+15=25>9, all hold so it forms a valid triangle.

Explanation

Option A: 5+6=11 which is less than 12, failing the inequality. Option C: 7+7=14, which equals the third side rather than strictly exceeding it. Option D: 8+8=16, also equal rather than strictly greater. All three cannot form a triangle. Option B: 9+10=19>15, 9+15=24>10, 10+15=25>9, all hold so it forms a valid triangle.

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15) Which set of side lengths forms an isosceles triangle?

4, 5, 10

7, 7, 10

9, 10, 12

8, 9, 10

An isosceles triangle requires at least two equal sides and must satisfy the triangle inequality. Option B has two equal sides of 7, and 7+7=14>10, so it is valid. Option A has no equal sides and 4+5=9 which is less than 10, failing the inequality. Options C and D have no equal sides so neither is isosceles.

Explanation

An isosceles triangle requires at least two equal sides and must satisfy the triangle inequality. Option B has two equal sides of 7, and 7+7=14>10, so it is valid. Option A has no equal sides and 4+5=9 which is less than 10, failing the inequality. Options C and D have no equal sides so neither is isosceles.

16) If two sides are 4 and 9, which range is correct for the third side x?

5 less than x less than 13

4 less than x less than 9

9 less than x less than 13

0 less than x less than 5

Lower bound: the absolute value of 9 minus 4 equals 5. Upper bound: 9+4=13. Therefore 5 less than x less than 13. Option B only spans the interval between the two given sides. Option C only covers the upper portion of the valid range. Option D gives values below the lower bound, all of which are invalid.

Explanation

Lower bound: the absolute value of 9 minus 4 equals 5. Upper bound: 9+4=13. Therefore 5 less than x less than 13. Option B only spans the interval between the two given sides. Option C only covers the upper portion of the valid range. Option D gives values below the lower bound, all of which are invalid.

17) Which of these is an impossible triangle?

7, 9, 15

6, 6, 12

5, 10, 14

8, 12, 18

Option B: 6+6=12, which equals the third side rather than being strictly greater. The strict triangle inequality requires the sum to exceed the third side, so this fails and no triangle exists. Option A: 7+9=16>15. Option C: 5+10=15>14. Option D: 8+12=20>18. Only option B fails.

Explanation

Option B: 6+6=12, which equals the third side rather than being strictly greater. The strict triangle inequality requires the sum to exceed the third side, so this fails and no triangle exists. Option A: 7+9=16>15. Option C: 5+10=15>14. Option D: 8+12=20>18. Only option B fails.

18) A triangle has sides 12 and 7. Which could be the third side?

5

16

20

22

Range: the absolute value of 12 minus 7 is less than x which is less than 12 plus 7, giving 5 less than x less than 19. Only 16 lies strictly between 5 and 19. Option A gives 5, which fails because x must be strictly greater than 5. Options C and D give 20 and 22, both exceeding the upper bound of 19.

Explanation

Range: the absolute value of 12 minus 7 is less than x which is less than 12 plus 7, giving 5 less than x less than 19. Only 16 lies strictly between 5 and 19. Option A gives 5, which fails because x must be strictly greater than 5. Options C and D give 20 and 22, both exceeding the upper bound of 19.

19) If two sides are 7 and 12, the third side x must satisfy which inequality?

5 less than x less than 19

5 less than or equal to x less than or equal to 19

7 less than x less than 12

0 less than x less than 19

The range uses the difference and sum of the given sides: the absolute value of 12 minus 7 is less than x which is less than 12 plus 7, giving 5 less than x less than 19. Both bounds are strict because equality produces a degenerate collinear case. Option B incorrectly uses non-strict inequalities. Option C only spans between the two given sides. Option D uses 0 as the lower bound, allowing invalid values.

Explanation

The range uses the difference and sum of the given sides: the absolute value of 12 minus 7 is less than x which is less than 12 plus 7, giving 5 less than x less than 19. Both bounds are strict because equality produces a degenerate collinear case. Option B incorrectly uses non-strict inequalities. Option C only spans between the two given sides. Option D uses 0 as the lower bound, allowing invalid values.

20) In any triangle with side lengths a, b, c, all three must hold: a+b greater than c, b+c greater than a, and c+a greater than b.

True

False

The answer is True. The Triangle Inequality Theorem requires all three strict inequalities to hold simultaneously. If even one fails, the three sides cannot close to form a triangle. The inequalities must all be strict because equality produces a degenerate case where the three points are collinear rather than forming a triangle.

Explanation

The answer is True. The Triangle Inequality Theorem requires all three strict inequalities to hold simultaneously. If even one fails, the three sides cannot close to form a triangle. The inequalities must all be strict because equality produces a degenerate case where the three points are collinear rather than forming a triangle.