Understanding Obtuse Triangles: How to Determine If a Triangle is Obtuse Using the Triangle Inequality Theorem

Regarding side length, a triangle is an obtuse triangle if …

A triangle is classified as an obtuse triangle if one of its angles is obtuse

A triangle is classified as an obtuse triangle if one of its angles is obtuse. An obtuse angle is an angle greater than 90 degrees but less than 180 degrees.

To determine if a triangle is obtuse, we need to examine the relationship between the side lengths of the triangle. In a triangle, the longest side, called the hypotenuse, is always opposite the largest angle. Therefore, if one of the side lengths is the longest, the opposite angle must be the largest angle.

Using the triangle inequality theorem, we can conclude that if the square of the longest side is greater than the sum of the squares of the other two sides, the triangle is obtuse.

In mathematical terms, let’s say the sides of a triangle are represented by a, b, and c. We need to check if:

a^2 > b^2 + c^2
b^2 > a^2 + c^2
c^2 > a^2 + b^2

If any of these conditions are true, then the triangle is obtuse.

For example, consider a triangle with side lengths a = 5, b = 4, and c = 3. We can check the conditions:

a^2 = 5^2 = 25
b^2 + c^2 = 4^2 + 3^2 = 16 + 9 = 25

Since a^2 is equal to the sum of b^2 and c^2, this triangle is not obtuse.

On the other hand, let’s consider a triangle with side lengths a = 6, b = 8, and c = 10:

a^2 = 6^2 = 36
b^2 + c^2 = 8^2 + 10^2 = 64 + 100 = 164

Since a^2 is less than the sum of b^2 and c^2, this triangle is obtuse.

Therefore, to determine if a triangle is an obtuse triangle, we need to compare the side lengths using the triangle inequality theorem. If the square of the longest side is greater than the sum of the squares of the other two sides, then the triangle is obtuse.

More Answers: