Mixed Applications of the Triangle Inequality

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

Equilateral

Isosceles Right

Scalene Obtuse

Scalene Right

A triangle with two equal sides is classified as isosceles. If it also contains a right angle, it is specifically known as an isosceles right triangle.

Explanation

A triangle with two equal sides is classified as isosceles. If it also contains a right angle, it is specifically known as an isosceles right triangle.

2) Which of these sets forms a valid triangle?

3, 3, 8

4, 5, 9

6, 10, 17

2, 2, 3

A valid triangle must satisfy the triangle inequality theorem, which states that the sum of the lengths of any two sides must be greater than the length of the third side. The set (2, 2, 3) meets this condition because 2 + 2 > 3, while the other sets do not.

Explanation

A valid triangle must satisfy the triangle inequality theorem, which states that the sum of the lengths of any two sides must be greater than the length of the third side. The set (2, 2, 3) meets this condition because 2 + 2 > 3, while the other sets do not.

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

Equilateral

Isosceles Right

Scalene Obtuse

Scalene Right

A triangle with two equal sides is classified as isosceles. If it also contains a right angle, it is specifically known as an isosceles right triangle.

Explanation

A triangle with two equal sides is classified as isosceles. If it also contains a right angle, it is specifically known as an isosceles right triangle.

4) Which of these sets forms a valid triangle?

3, 3, 8

4, 5, 9

6, 10, 17

2, 2, 3

A valid triangle must satisfy the triangle inequality theorem, which states that the sum of the lengths of any two sides must be greater than the length of the third side. The set (2, 2, 3) meets this condition because 2 + 2 > 3, while the other sets do not.

Explanation

A valid triangle must satisfy the triangle inequality theorem, which states that the sum of the lengths of any two sides must be greater than the length of the third side. The set (2, 2, 3) meets this condition because 2 + 2 > 3, while the other sets do not.

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

Equilateral

Isosceles Right

Scalene Obtuse

Scalene Right

A triangle with two equal sides is classified as isosceles. If it also contains a right angle, it is specifically known as an isosceles right triangle.

Explanation

A triangle with two equal sides is classified as isosceles. If it also contains a right angle, it is specifically known as an isosceles right triangle.

6) Which of these sets can form a triangle?

2, 3, 6

4, 7, 12

5, 9, 13

8, 10, 19

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. The set 5, 9, and 13 satisfies this condition, while the others do not.

Explanation

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. The set 5, 9, and 13 satisfies this condition, while the others do not.

7) Which of these sets can form a triangle?

2, 3, 6

4, 7, 12

5, 9, 13

8, 10, 19

A set of three lengths can form a triangle if the sum of any two lengths is greater than the third length. In this case, the set (5, 9, 13) satisfies this condition, while the other sets do not.

Explanation

A set of three lengths can form a triangle if the sum of any two lengths is greater than the third length. In this case, the set (5, 9, 13) satisfies this condition, while the other sets do not.

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

5

16

20

22

In a triangle, the length of any side must be less than the sum of the other two sides and greater than the absolute difference of the other two sides. Here, 12 + 7 = 19 (sum) and 12 - 7 = 5 (difference). The third side must be greater than 5 and less than 19. The option 16 meets this condition, while the others do not.

Explanation

In a triangle, the length of any side must be less than the sum of the other two sides and greater than the absolute difference of the other two sides. Here, 12 + 7 = 19 (sum) and 12 - 7 = 5 (difference). The third side must be greater than 5 and less than 19. The option 16 meets this condition, while the others do not.

9) Which of these is an impossible triangle?

7, 9, 15

6, 6, 12

5, 10, 14

8, 12, 18

An impossible triangle is formed when the sum of the lengths of any two sides is less than or equal to the length of the third side. In option B, the sum of the two equal sides (6 + 6 = 12) is equal to the length of the third side (12), which violates the triangle inequality theorem, making it impossible to form a triangle.

Explanation

An impossible triangle is formed when the sum of the lengths of any two sides is less than or equal to the length of the third side. In option B, the sum of the two equal sides (6 + 6 = 12) is equal to the length of the third side (12), which violates the triangle inequality theorem, making it impossible to form a triangle.

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

5

16

20

22

In a triangle, the length of any side must be less than the sum of the other two sides and greater than the absolute difference of the other two sides. Here, 12 + 7 = 19 (sum) and 12 - 7 = 5 (difference). The third side must be greater than 5 and less than 19. The option 16 meets this condition, while the others do not.

Explanation

In a triangle, the length of any side must be less than the sum of the other two sides and greater than the absolute difference of the other two sides. Here, 12 + 7 = 19 (sum) and 12 - 7 = 5 (difference). The third side must be greater than 5 and less than 19. The option 16 meets this condition, while the others do not.

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

Scalene

Isosceles

Right

Obtuse

A triangle with two equal sides is defined as an isosceles triangle. This type of triangle has at least two sides that are of equal length, which directly corresponds to the property stated in the question.

Explanation

A triangle with two equal sides is defined as an isosceles triangle. This type of triangle has at least two sides that are of equal length, which directly corresponds to the property stated in the question.

12) Which of these sets cannot form a triangle?

5, 6, 12

8, 8, 8

9, 10, 15

7, 7, 12

A set of lengths can form a triangle if the sum of the lengths of any two sides is greater than the length of the remaining side. In set A (5, 6, 12), 5 + 6 is not greater than 12, so it cannot form a triangle.

Explanation

A set of lengths can form a triangle if the sum of the lengths of any two sides is greater than the length of the remaining side. In set A (5, 6, 12), 5 + 6 is not greater than 12, so it cannot form a triangle.

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

5

16

20

22

In a triangle, the length of any side must be less than the sum of the other two sides and greater than the absolute difference of the other two sides. Here, 12 + 7 = 19 (sum) and 12 - 7 = 5 (difference). The third side must be greater than 5 and less than 19. The option 16 meets this condition, while the others do not.

Explanation

In a triangle, the length of any side must be less than the sum of the other two sides and greater than the absolute difference of the other two sides. Here, 12 + 7 = 19 (sum) and 12 - 7 = 5 (difference). The third side must be greater than 5 and less than 19. The option 16 meets this condition, while the others do not.

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

5

16

20

22

In a triangle, the length of any side must be less than the sum of the other two sides and greater than the absolute difference of the other two sides. Here, 12 + 7 = 19 (sum) and 12 - 7 = 5 (difference). The third side must be greater than 5 and less than 19. The option 16 meets this condition, while the others do not.

Explanation

In a triangle, the length of any side must be less than the sum of the other two sides and greater than the absolute difference of the other two sides. Here, 12 + 7 = 19 (sum) and 12 - 7 = 5 (difference). The third side must be greater than 5 and less than 19. The option 16 meets this condition, while the others do not.

15) Which set of side lengths could form a right triangle?

6, 8, 10

7, 8, 12

5, 5, 9

4, 5, 10

A right triangle can be formed if the squares of the two shorter sides add up to the square of the longest side. In this case, 6² + 8² = 36 + 64 = 100, which equals 10² = 100. The other sets do not satisfy this condition.

Explanation

A right triangle can be formed if the squares of the two shorter sides add up to the square of the longest side. In this case, 6² + 8² = 36 + 64 = 100, which equals 10² = 100. The other sets do not satisfy this condition.

16) If a triangle has sides 10 and 4, which of the following is possible for the third side?

4

5

6

12

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, the third side must be less than 10 + 4 (which is 14) and greater than 10 - 4 (which is 6). Therefore, a length of 12 is valid, while 4, 5, and 6 do not satisfy the conditions.

Explanation

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, the third side must be less than 10 + 4 (which is 14) and greater than 10 - 4 (which is 6). Therefore, a length of 12 is valid, while 4, 5, and 6 do not satisfy the conditions.

17) Which of these is always true about a triangle?

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

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

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

The longest side can be any number:

This statement is a reflection of the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side.

Explanation

This statement is a reflection of the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side.

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

3 < x < 21

0 < x < 9

12 < x < 21

X > 21

In any triangle, the sum of the lengths of any two sides must be greater than the length of the third side. For sides 9 and 12, x must be greater than 3 (9 - 12) and less than 21 (9 + 12). Hence, the correct inequality is 3

Explanation

In any triangle, the sum of the lengths of any two sides must be greater than the length of the third side. For sides 9 and 12, x must be greater than 3 (9 - 12) and less than 21 (9 + 12). Hence, the correct inequality is 3

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

Equilateral

Isosceles Right

Scalene Obtuse

Scalene Right

A triangle with two equal sides is classified as isosceles. If it also contains a right angle, it is specifically known as an isosceles right triangle.

Explanation

A triangle with two equal sides is classified as isosceles. If it also contains a right angle, it is specifically known as an isosceles right triangle.

20) Which of these sets forms a valid triangle?

3, 3, 8

4, 5, 9

6, 10, 17

2, 2, 3

A valid triangle must satisfy the triangle inequality theorem, which states that the sum of the lengths of any two sides must be greater than the length of the third side. The set (2, 2, 3) meets this condition because 2 + 2 > 3, while the other sets do not.

Explanation

A valid triangle must satisfy the triangle inequality theorem, which states that the sum of the lengths of any two sides must be greater than the length of the third side. The set (2, 2, 3) meets this condition because 2 + 2 > 3, while the other sets do not.