Triangle Inequality Quiz: Master Triangle Inequality Quiz

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

2, 3, 2006

4, 7, 12

5, 9, 13

8, 10, 0194

Check triangle inequality for each set: (i) 2 + 3 = 5 ≤ 6 ⇒ not a triangle; (ii) 4 + 7 = 11 < 12 ⇒ not a triangle; (iii) 5 + 9 = 14 > 13, 5 + 13 = 18 > 9, 9 + 13 = 22 > 5 ⇒ valid; (iv) 8 + 10 = 18 < 194 ⇒ not a triangle. So 5, 9, 13 works.

Explanation

Check triangle inequality for each set: (i) 2 + 3 = 5 ≤ 6 ⇒ not a triangle; (ii) 4 + 7 = 11 < 12 ⇒ not a triangle; (iii) 5 + 9 = 14 > 13, 5 + 13 = 18 > 9, 9 + 13 = 22 > 5 ⇒ valid; (iv) 8 + 10 = 18 < 194 ⇒ not a triangle. So 5, 9, 13 works.

2) In any triangle with side lengths a, b, c, all three must hold: a + b > c, b + c > a, and c + a > b.

True

False

Triangle Inequality Theorem requires each side to be less than the sum of the other two. All three strict inequalities must be true simultaneously; if any fails, no triangle exists.

Explanation

Triangle Inequality Theorem requires each side to be less than the sum of the other two. All three strict inequalities must be true simultaneously; if any fails, no triangle exists.

Submit

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

5

16

20

22

Range: |12 − 7| < x < 12 + 7 ⇒ 5 < x < 19. Only 16 lies strictly between 5 and 19; 5 fails (not strictly greater), 20 and 22 exceed 19.

Explanation

Range: |12 − 7| < x < 12 + 7 ⇒ 5 < x < 19. Only 16 lies strictly between 5 and 19; 5 fails (not strictly greater), 20 and 22 exceed 19.

5) Which of these is an impossible triangle?

7, 9, 15

6, 6, 12

5, 10, 14

8, 12, 18

Check each: (i) 7 + 9 = 16 > 15 ⇒ ok; (ii) 6 + 6 = 12 (equals) ⇒ fails strict inequality ⇒ impossible; (iii) 5 + 10 = 15 > 14, others also ok; (iv) 8 + 12 = 20 > 18, others ok.

Explanation

Check each: (i) 7 + 9 = 16 > 15 ⇒ ok; (ii) 6 + 6 = 12 (equals) ⇒ fails strict inequality ⇒ impossible; (iii) 5 + 10 = 15 > 14, others also ok; (iv) 8 + 12 = 20 > 18, others ok.

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

5 < x < 13

4 < x < 9

9 < x < 13

0 < x < 5

Use |9 − 4|

Explanation

Use |9 − 4|

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

4, 5, 2006

7, 7, 10

9, 10, 12

8, 9, 10

Isosceles requires at least two equal sides and must satisfy triangle inequality. 7, 7, 10 has two equal sides and 7 + 7 = 14 > 10, so valid.

Explanation

Isosceles requires at least two equal sides and must satisfy triangle inequality. 7, 7, 10 has two equal sides and 7 + 7 = 14 > 10, so valid.

8) Select all sets that CANNOT form a triangle.

5, 6, 12

9, 10, 15

7, 7, 14

3, 4, 2008

8, 8, 16

Test sums: (A) 5 + 6 = 11 15 and other sums ok ⇒ valid; (C) 7 + 7 = 14 (equals) ⇒ cannot; (D) 3 + 4 = 7

Explanation

Test sums: (A) 5 + 6 = 11 15 and other sums ok ⇒ valid; (C) 7 + 7 = 14 (equals) ⇒ cannot; (D) 3 + 4 = 7

Submit

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

4

5

6

12

Range: |10 − 4|

Explanation

Range: |10 − 4|

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

True

False

Triangle inequality requires each side to be strictly less than the sum of the other two. Equality (e.g., a = b + c) forms a straight line, not a triangle.

Explanation

Triangle inequality requires each side to be strictly less than the sum of the other two. Equality (e.g., a = b + c) forms a straight line, not a triangle.

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

1 < x < 14

2 < x < 14

2 ≤ x < 14

2 < x ≤ 14

Use |8 − 6|

Explanation

Use |8 − 6|

12) Which set could form a right triangle?

6, 8, 10

7, 8, 12

5, 5, 12

4, 5, 10

Check Pythagorean and triangle inequality: 6² + 8² = 36 + 64 = 100 = 10² ⇒ right triangle; also 6 + 8 > 10, etc. Others are not Pythagorean triples.

Explanation

Check Pythagorean and triangle inequality: 6² + 8² = 36 + 64 = 100 = 10² ⇒ right triangle; also 6 + 8 > 10, etc. Others are not Pythagorean triples.

13) With sides 11 and 7, the largest possible integer length of the third side is ____.

Upper bound is 11 + 7 = 18, but x must be strictly less than 18, so the greatest integer is 17.

Explanation

Upper bound is 11 + 7 = 18, but x must be strictly less than 18, so the greatest integer is 17.

Submit

14) With sides 11 and 7, the smallest possible integer length of the third side is ____.

Lower bound is |11 − 7| = 4, but x must be strictly greater than 4, so the smallest integer exceeding 4 is 5.

Explanation

Lower bound is |11 − 7| = 4, but x must be strictly greater than 4, so the smallest integer exceeding 4 is 5.

Submit

15) Select all sets that CAN form a triangle.

2, 2, 2005

6, 8, 2009

7, 24, 25

9, 11, 15

3, 4, 2007

Check each: (A) 2 + 2 = 4 25 and others ok ⇒ can; (D) 9 + 11 = 20 > 15 and others ok ⇒ can; (E) 3 + 4 = 7 (equals) ⇒ cannot.

Explanation

Check each: (A) 2 + 2 = 4 25 and others ok ⇒ can; (D) 9 + 11 = 20 > 15 and others ok ⇒ can; (E) 3 + 4 = 7 (equals) ⇒ cannot.

Submit

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

12

11

0

13

Range: |6 − 6|

Explanation

Range: |6 − 6|

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

True

False

Let c be the longest side. Triangle inequality requires a + b > c. Since c ≥ a and c ≥ b, the strict inequality ensures c is less than the sum of the other two.

Explanation

Let c be the longest side. Triangle inequality requires a + b > c. Since c ≥ a and c ≥ b, the strict inequality ensures c is less than the sum of the other two.