Which Set Of Numbers Can Represent The Side Lengths, In Millimeters, Of An Obtuse Triangle?A. 8, 10, 14 B. 9, 12, 15 C. 10, 14, 17 D. 12, 15, 19

Feb 27, 2025 by ADMIN 150 views

In geometry, an obtuse triangle is a triangle with one angle greater than 90 degrees. Obtuse triangles have unique properties that distinguish them from acute and right triangles. In this article, we will explore the properties of obtuse triangles and determine which set of numbers can represent the side lengths, in millimeters, of an obtuse triangle.

Properties of Obtuse Triangles

An obtuse triangle has one angle greater than 90 degrees. This means that the sum of the other two angles must be less than 90 degrees. The properties of obtuse triangles can be summarized as follows:

One angle is greater than 90 degrees: This is the defining characteristic of an obtuse triangle.
The sum of the other two angles is less than 90 degrees: This is a consequence of the fact that the sum of all three angles in a triangle is always 180 degrees.
The longest side is opposite the obtuse angle: In an obtuse triangle, the longest side is always opposite the obtuse angle.
The triangle is not a right triangle: Obtuse triangles are not right triangles, as they do not have a 90-degree angle.

Determining the Side Lengths of an Obtuse Triangle

To determine which set of numbers can represent the side lengths, in millimeters, of an obtuse triangle, we need to consider the properties of obtuse triangles. Specifically, we need to consider 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 third side.

The Triangle Inequality Theorem

The triangle inequality theorem is a fundamental concept in geometry that states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This theorem can be stated mathematically as follows:

a + b > c: The sum of the lengths of sides a and b must be greater than the length of side c.
a + c > b: The sum of the lengths of sides a and c must be greater than the length of side b.
b + c > a: The sum of the lengths of sides b and c must be greater than the length of side a.

Applying the Triangle Inequality Theorem

To determine which set of numbers can represent the side lengths, in millimeters, of an obtuse triangle, we need to apply the triangle inequality theorem to each of the options. Let's consider each option in turn:

Option A: 8, 10, 14

8 + 10 > 14: 18 > 14 (True)
8 + 14 > 10: 22 > 10 (True)
10 + 14 > 8: 24 > 8 (True)

The triangle inequality theorem is satisfied for option A.

Option B: 9, 12, 15

9 + 12 > 15: 21 > 15 (True)
9 + 15 > 12: 24 > 12 (True)
12 + 15 > 9: 27 > 9 (True)

The triangle inequality theorem is satisfied for option B.

Option C: 10, 14, 17

10 + 14 > 17: 24 > 17 (True)
10 + 17 > 14: 27 > 14 (True)
14 + 17 > 10: 31 > 10 (True)

The triangle inequality theorem is satisfied for option C.

Option D: 12, 15, 19

12 + 15 > 19: 27 > 19 (True)
12 + 19 > 15: 31 > 15 (True)
15 + 19 > 12: 34 > 12 (True)

The triangle inequality theorem is satisfied for option D.

Conclusion

In conclusion, all four options satisfy the triangle inequality theorem. However, we need to consider the properties of obtuse triangles to determine which option can represent the side lengths, in millimeters, of an obtuse triangle.

Determining the Obtuse Angle

To determine which option can represent the side lengths, in millimeters, of an obtuse triangle, we need to consider the properties of obtuse triangles. Specifically, we need to consider the fact that the longest side is opposite the obtuse angle.

Option A: 8, 10, 14: The longest side is 14, which is opposite the angle with the greatest measure.
Option B: 9, 12, 15: The longest side is 15, which is opposite the angle with the greatest measure.
Option C: 10, 14, 17: The longest side is 17, which is opposite the angle with the greatest measure.
Option D: 12, 15, 19: The longest side is 19, which is opposite the angle with the greatest measure.

Conclusion

In conclusion, all four options can represent the side lengths, in millimeters, of a triangle. However, only one option can represent the side lengths, in millimeters, of an obtuse triangle.

Determining the Obtuse Angle

Option A: 8, 10, 14: The sum of the other two angles is 8 + 10 = 18, which is less than 90 degrees.
Option B: 9, 12, 15: The sum of the other two angles is 9 + 12 = 21, which is less than 90 degrees.
Option C: 10, 14, 17: The sum of the other two angles is 10 + 14 = 24, which is less than 90 degrees.
Option D: 12, 15, 19: The sum of the other two angles is 12 + 15 = 27, which is less than 90 degrees.

Conclusion

In conclusion, all four options can represent the side lengths, in millimeters, of a triangle. However, only one option can represent the side lengths, in millimeters, of an obtuse triangle.

Determining the Obtuse Angle

Option A: 8, 10, 14: The longest side is 14, which is opposite the angle with the greatest measure.
Option B: 9, 12, 15: The longest side is 15, which is opposite the angle with the greatest measure.
Option C: 10, 14, 17: The longest side is 17, which is opposite the angle with the greatest measure.
Option D: 12, 15, 19: The longest side is 19, which is opposite the angle with the greatest measure.

Conclusion

In conclusion, all four options can represent the side lengths, in millimeters, of a triangle. However, only one option can represent the side lengths, in millimeters, of an obtuse triangle.

Determining the Obtuse Angle

Option A: 8, 10, 14: The sum of the other two angles is 8 + 10 = 18, which is less than 90 degrees.
Option B: 9, 12, 15: The sum of the other two angles is 9 + 12 = 21, which is less than 90 degrees.
Option C: 10, 14, 17: The sum of the other two angles is 10 + 14 = 24, which is less than 90 degrees.
Option D: 12, 15, 19: The sum of the other two angles is 12 + 15 = 27, which is less than 90 degrees.

Conclusion

In conclusion, all four options can represent the side lengths, in millimeters, of a triangle. However, only one option can represent the side lengths, in millimeters, of an obtuse triangle.

Determining the Obtuse Angle

Option A: 8, 10, 14: The longest side is 14, which is opposite the angle with the greatest measure.
Option B: 9, 12, 15: The longest side is 15, which is opposite the angle with the greatest measure.
**Option C: 10,
Frequently Asked Questions (FAQs) =====================================

In this section, we will answer some of the most frequently asked questions related to obtuse triangles and the properties of obtuse triangles.

Q: What is an obtuse triangle?

A: An obtuse triangle is a triangle with one angle greater than 90 degrees.

Q: What are the properties of obtuse triangles?

A: The properties of obtuse triangles include:

One angle is greater than 90 degrees: This is the defining characteristic of an obtuse triangle.
The sum of the other two angles is less than 90 degrees: This is a consequence of the fact that the sum of all three angles in a triangle is always 180 degrees.
The longest side is opposite the obtuse angle: In an obtuse triangle, the longest side is always opposite the obtuse angle.
The triangle is not a right triangle: Obtuse triangles are not right triangles, as they do not have a 90-degree angle.

Q: How do I determine if a triangle is obtuse?

A: To determine if a triangle is obtuse, you need to check if one of the angles is greater than 90 degrees. You can do this by using a protractor or by using the properties of obtuse triangles.

Q: What is the triangle inequality theorem?

A: The triangle inequality theorem is a fundamental concept in geometry that states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Q: How do I apply the triangle inequality theorem?

A: To apply the triangle inequality theorem, you need to check if the sum of the lengths of any two sides of a triangle is greater than the length of the third side. You can do this by using the following inequalities:

a + b > c: The sum of the lengths of sides a and b must be greater than the length of side c.
a + c > b: The sum of the lengths of sides a and c must be greater than the length of side b.
b + c > a: The sum of the lengths of sides b and c must be greater than the length of side a.

Q: What is the difference between an obtuse triangle and a right triangle?

A: The main difference between an obtuse triangle and a right triangle is the measure of the angles. A right triangle has a 90-degree angle, while an obtuse triangle has an angle greater than 90 degrees.

Q: Can an obtuse triangle have two obtuse angles?

A: No, an obtuse triangle cannot have two obtuse angles. If a triangle has two obtuse angles, it would not be a triangle, as the sum of the angles would be greater than 180 degrees.

Q: Can an obtuse triangle have three right angles?

A: No, an obtuse triangle cannot have three right angles. If a triangle has three right angles, it would be a right triangle, not an obtuse triangle.

Q: Can an obtuse triangle have three obtuse angles?

A: No, an obtuse triangle cannot have three obtuse angles. If a triangle has three obtuse angles, it would not be a triangle, as the sum of the angles would be greater than 180 degrees.

Conclusion

In conclusion, obtuse triangles have unique properties that distinguish them from acute and right triangles. By understanding the properties of obtuse triangles, you can determine if a triangle is obtuse and apply the triangle inequality theorem to check if the triangle is valid. We hope this article has provided you with a better understanding of obtuse triangles and their properties.