Understanding Obtuse Triangles

Oct 13, 2025 by leolavanderia.com.br 17 views

Understanding Obtuse Triangles

An obtuse triangle is a type of triangle where one of the angles is greater than 90 degrees. In order to determine if a set of three numbers represents the sides of an obtuse triangle, we need to use the triangle inequality theorem and the properties of obtuse triangles.

Triangle Inequality Theorem

The triangle inequality theorem 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 is essential in determining if a set of three numbers can form a triangle.

Properties of Obtuse Triangles

An obtuse triangle has one angle greater than 90 degrees. This means that the longest side of the triangle is opposite the obtuse angle. In order to determine if a set of three numbers represents the sides of an obtuse triangle, we need to check if the longest side is opposite the largest angle.

Checking the Sets of Numbers

Let's check each set of numbers to see if they represent the sides of an obtuse triangle.

$4, 7, 8$

To check if this set of numbers represents the sides of an obtuse triangle, we need to use the triangle inequality theorem. The sum of the lengths of any two sides must be greater than the length of the third side.

$4 + 7 > 8$ (True)
$7 + 8 > 4$ (True)
$4 + 8 > 7$ (True)

Since all three inequalities are true, this set of numbers represents the sides of a triangle. However, we need to check if the longest side is opposite the largest angle.

The longest side is 8.
The largest angle is opposite the longest side, which is 8.

Since the longest side is opposite the largest angle, this set of numbers represents the sides of an obtuse triangle.

$3, 4, 5$

To check if this set of numbers represents the sides of an obtuse triangle, we need to use the triangle inequality theorem.

$3 + 4 > 5$ (False)
$4 + 5 > 3$ (True)
$3 + 5 > 4$ (True)

Since not all three inequalities are true, this set of numbers does not represent the sides of a triangle.

$2, 2, 3$

To check if this set of numbers represents the sides of an obtuse triangle, we need to use the triangle inequality theorem.

$2 + 2 > 3$ (False)
$2 + 3 > 2$ (True)
$2 + 3 > 2$ (True)

Since not all three inequalities are true, this set of numbers does not represent the sides of a triangle.

$6, 8, 9$

To check if this set of numbers represents the sides of an obtuse triangle, we need to use the triangle inequality theorem.

$6 + 8 > 9$ (True)
$8 + 9 > 6$ (True)
$6 + 9 > 8$ (True)

Since all three inequalities are true, this set of numbers represents the sides of a triangle. However, we need to check if the longest side is opposite the largest angle.

The longest side is 9.
The largest angle is opposite the longest side, which is 9.

Since the longest side is opposite the largest angle, this set of numbers represents the sides of an obtuse triangle.

$3, 5, 6$

To check if this set of numbers represents the sides of an obtuse triangle, we need to use the triangle inequality theorem.

$3 + 5 > 6$ (True)
$5 + 6 > 3$ (True)
$3 + 6 > 5$ (True)

Since all three inequalities are true, this set of numbers represents the sides of a triangle. However, we need to check if the longest side is opposite the largest angle.

The longest side is 6.
The largest angle is opposite the longest side, which is 6.

Since the longest side is opposite the largest angle, this set of numbers represents the sides of an obtuse triangle.

Conclusion

In conclusion, the sets of numbers that represent the sides of an obtuse triangle are:

$4, 7, 8$
$6, 8, 9$
$3, 5, 6$

Q: What is an obtuse triangle?

A: An obtuse triangle is a type of triangle where one of the angles is greater than 90 degrees.

Q: How do I determine if a set of three numbers represents the sides of an obtuse triangle?

A: To determine if a set of three numbers represents the sides of an obtuse triangle, you need to use the triangle inequality theorem and check if the longest side is opposite the largest angle.

Q: What is the triangle inequality theorem?

A: The triangle inequality theorem 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 check if the longest side is opposite the largest angle?

A: To check if the longest side is opposite the largest angle, you need to compare the lengths of the sides and the angles of the triangle.

Q: What are some examples of obtuse triangles?

A: Some examples of obtuse triangles include:

A triangle with sides 4, 7, and 8
A triangle with sides 6, 8, and 9
A triangle with sides 3, 5, and 6

Q: Can a right triangle be an obtuse triangle?

A: No, a right triangle cannot be an obtuse triangle. A right triangle has one angle that is exactly 90 degrees, while an obtuse triangle has one angle that is greater than 90 degrees.

Q: Can an equilateral triangle be an obtuse triangle?

A: No, an equilateral triangle cannot be an obtuse triangle. An equilateral triangle has all sides of equal length, while an obtuse triangle has one side that is longer than the other two.

Q: Can a scalene triangle be an obtuse triangle?

A: Yes, a scalene triangle can be an obtuse triangle. A scalene triangle has all sides of different lengths, and it can have one angle that is greater than 90 degrees.

Q: How do I find the measure of an obtuse angle in a triangle?

A: To find the measure of an obtuse angle in a triangle, you need to use the fact that the sum of the measures of the angles in a triangle is 180 degrees. Since one angle is greater than 90 degrees, you can subtract the measures of the other two angles from 180 degrees to find the measure of the obtuse angle.

Q: Can an obtuse triangle have two obtuse angles?

A: No, an obtuse triangle cannot have two obtuse angles. Since the sum of the measures of the angles in a triangle is 180 degrees, if one angle is greater than 90 degrees, the other two angles must be acute (less than 90 degrees).

Q: Can an obtuse triangle have three obtuse angles?

A: No, an obtuse triangle cannot have three obtuse angles. Since the sum of the measures of the angles in a triangle is 180 degrees, if all three angles are greater than 90 degrees, the sum would be greater than 180 degrees, which is not possible.

Conclusion

In conclusion, obtuse triangles are a type of triangle where one of the angles is greater than 90 degrees. To determine if a set of three numbers represents the sides of an obtuse triangle, you need to use the triangle inequality theorem and check if the longest side is opposite the largest angle.