Mixed Applications of the Triangle Inequality

  • 7th Grade
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| Attempts: 11 | Questions: 20 | Updated: Dec 2, 2025
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1) Which of these sets can form 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.

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About This Quiz
Mixed Applications Of The Triangle Inequality - Quiz

Want to see how one simple rule can solve many triangle questions? This quiz will challenge you with all kinds of problems — from checking if three sides can fit together, to finding ranges, to identifying special triangles like isosceles or right. You’ll practice the triangle inequality in different ways... see moreand grow more confident with each problem!
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2) Which inequality must be true for a triangle with sides a, b, and c?

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.

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

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.

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4) Which of these is an impossible triangle?

Explanation

6 + 6 = 12 equals the third side.

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

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

Explanation

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

9 − 4 = 5, 9 + 4 = 13 → ✅ 5 < x < 13.

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

Explanation

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

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7) Which of these sets cannot form a triangle?

Explanation

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

That breaks the triangle rule.

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

Explanation

The triangle inequality rule states:

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

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

Explanation

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

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

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10) Which set could form a right triangle?

Explanation

Check with the Pythagorean theorem:

6² + 8² = 36 + 64 = 100 = 10².

That’s a perfect right triangle.

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

Explanation

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

12 fits nicely in that range.

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12) Which is always true about a triangle?

Explanation

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

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

Explanation

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

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14) Which set cannot form a triangle?

Explanation

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

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

Explanation

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

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16) Which of the following is a scalene triangle?

Explanation

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

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

Explanation

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

11 works.

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

Explanation

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

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19) Which of these sets forms a valid triangle?

Explanation

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

This set forms a valid, small isosceles triangle.

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

Explanation

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

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  • Answered
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Which of these sets can form a triangle?
Which inequality must be true for a triangle with sides a, b, and c?
A triangle has sides 12 and 7. Which could be the third side?
Which of these is an impossible triangle?
If two sides are 4 and 9, which of the following ranges is correct for...
A triangle has two equal sides. What type of triangle must it be?
Which of these sets cannot form a triangle?
The triangle inequality guarantees that the sum of the lengths of any...
Which set of side lengths forms an isosceles triangle?
Which set could form a right triangle?
If a triangle has sides 10 and 4, which third side works?
Which is always true about a triangle?
A triangle has sides 9, 12, and x. Which inequality must hold?
Which set cannot form a triangle?
Two sides are 15 and 8. Which is a valid third side?
Which of the following is a scalene triangle?
A triangle has sides 6 and 6. Which could be the third side?
Which is the correct description of the triangle inequality?
Which of these sets forms a valid triangle?
If a triangle has two equal sides and a right angle, what type is it?
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