Triangle Inequality: Mixed Applications

1) Which of these sets can form a triangle?

2, 3, 6

4, 7, 12

5, 9, 13

8, 10, 194

To form a triangle, the sum of any two sides must be greater than the third.

Here, 5 + 9 = 14 > 13, 9 + 13 = 22 > 5, and 5 + 13 = 18 > 9.

✅ All conditions are met — this set forms a triangle.

Explanation

To form a triangle, the sum of any two sides must be greater than the third.

Here, 5 + 9 = 14 > 13, 9 + 13 = 22 > 5, and 5 + 13 = 18 > 9.

✅ All conditions are met — this set forms a triangle.

2) Which inequality must be true for a triangle with sides a, b, and c?

A + b = c

A + b > c

A + b < c

A + b ≥ c

In any triangle, the sum of any two sides must be greater than the third.

If it’s equal, the shape becomes a straight line, not a triangle.

Explanation

In any triangle, the sum of any two sides must be greater than the third.

If it’s equal, the shape becomes a straight line, not a triangle.

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

5

16

20

22

The third side must be between the difference and sum of the other two sides.

12 − 7 = 5 and 12 + 7 = 19 → so 5 < x < 19.

✅ 16 fits perfectly in this range.

Explanation

The third side must be between the difference and sum of the other two sides.

12 − 7 = 5 and 12 + 7 = 19 → so 5 < x < 19.

✅ 16 fits perfectly in this range.

4) Which of these is an impossible triangle?

7, 9, 15

6, 6, 12

5, 10, 14

8, 12, 18

6 + 6 = 12 equals the third side.

When the sum equals the longest side, it’s a straight line, not a triangle.

Explanation

6 + 6 = 12 equals the third side.

When the sum equals the longest side, it’s a straight line, not a triangle.

5) If two sides are 4 and 9, which of the following ranges is correct for the third side?

5 < x < 13

4 < x < 9

9 < x < 13

0 < x < 5

The third side must be greater than the difference and less than the sum.

9 − 4 = 5, 9 + 4 = 13 → ✅ 5 < x < 13.

Explanation

The third side must be greater than the difference and less than the sum.

9 − 4 = 5, 9 + 4 = 13 → ✅ 5 < x < 13.

6) A triangle has two equal sides. What type of triangle must it be?

Scalene

Isosceles

Obtuse

When two sides are equal, the triangle is isosceles, and two angles are also equal.

Explanation

When two sides are equal, the triangle is isosceles, and two angles are also equal.

7) Which of these sets cannot form a triangle?

5, 6, 12

8, 8, 2008

9, 10, 15

7, 7, 12

Add the two smaller sides: 5 + 6 = 11, which is less than 12.

That breaks the triangle rule.

Explanation

Add the two smaller sides: 5 + 6 = 11, which is less than 12.

That breaks the triangle rule.

8) The triangle inequality guarantees that the sum of the lengths of any two sides is:

Less than the third side

Greater than the third side

Equal to the third side

Double the third side

The triangle inequality rule states:

The sum of any two sides must be greater than the third side.

Explanation

The triangle inequality rule states:

The sum of any two sides must be greater than the third side.

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

4, 5, 6

7, 7, 10

9, 10, 12

8, 9, 10

Two equal sides (7 and 7) make it isosceles.

Also, 7 + 7 > 10 satisfies the triangle condition.

Explanation

Two equal sides (7 and 7) make it isosceles.

Also, 7 + 7 > 10 satisfies the triangle condition.

10) Which set could form a right triangle?

6, 8, 10

7, 8, 12

5, 5, 2009

4, 5, 10

Check with the Pythagorean theorem:

6² + 8² = 36 + 64 = 100 = 10².

That’s a perfect right triangle.

Explanation

Check with the Pythagorean theorem:

6² + 8² = 36 + 64 = 100 = 10².

That’s a perfect right triangle.

11) If a triangle has sides 10 and 4, which third side works?

4

5

6

12

The third side must be between 10 − 4 = 6 and 10 + 4 = 14.

12 fits nicely in that range.

Explanation

The third side must be between 10 − 4 = 6 and 10 + 4 = 14.

12 fits nicely in that range.

12) Which is always true about a triangle?

The longest side is always less than the sum of the other two sides.

The longest side is greater than the sum of the other two sides.

The longest side is equal to the sum of the other two sides.

The longest side can be any number.

The triangle inequality ensures that the longest side is always smaller than the sum of the other two.

Explanation

The triangle inequality ensures that the longest side is always smaller than the sum of the other two.

13) A triangle has sides 9, 12, and x. Which inequality must hold?

3 < x < 21

0 < x < 9

12 < x < 21

X > 21

Subtract and add: 12 − 9 = 3, 12 + 9 = 21 → 3

Explanation

Subtract and add: 12 − 9 = 3, 12 + 9 = 21 → 3

14) Which set cannot form a triangle?

2, 2, 5

6, 8, 2009

7, 24, 25

9, 11, 15

2 + 2 = 4, which is less than 5, so it cannot form a triangle.

Explanation

2 + 2 = 4, which is less than 5, so it cannot form a triangle.

15) Two sides are 15 and 8. Which is a valid third side?

6

7

20

24

15 − 8 = 7 and 15 + 8 = 23, so 7
20 fits this range.

Explanation

15 − 8 = 7 and 15 + 8 = 23, so 7
20 fits this range.

16) Which of the following is a scalene triangle?

5, 5, 5

7, 7, 10

9, 10, 11

8, 8, 12

All sides are different, so it’s a scalene triangle.

Explanation

All sides are different, so it’s a scalene triangle.

17) A triangle has sides 6 and 6. Which could be the third side?

12

11

0

13

For sides 6 and 6, the third side must be less than 12 and greater than 0.

11 works.

Explanation

For sides 6 and 6, the third side must be less than 12 and greater than 0.

11 works.

18) Which is the correct description of the triangle inequality?

Any side must be less than the sum of the other two.

Any side must be greater than the sum of the other two.

Any side must equal the sum of the other two.

Any side must be less than the product of the other two.

That’s the triangle inequality rule — any one side is always less than the sum of the other two.

Explanation

That’s the triangle inequality rule — any one side is always less than the sum of the other two.

19) Which of these sets forms a valid triangle?

3, 3, 8

4, 5, 2009

6, 10, 17

2, 2, 2003

2 + 2 = 4 > 3, and all other pairs also work.

This set forms a valid, small isosceles triangle.

Explanation

2 + 2 = 4 > 3, and all other pairs also work.

This set forms a valid, small isosceles triangle.

20) If a triangle has two equal sides and a right angle, what type is it?

Equilateral

Isosceles Right

Scalene Obtuse

Scalene Right

Two equal sides plus one 90° angle make it an isosceles right triangle.

Explanation

Two equal sides plus one 90° angle make it an isosceles right triangle.

Mixed Applications of the Triangle Inequality