How to Identify an Obtuse Triangle: Using the Law of Cosines to Determine Triangle Types

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

A triangle is considered an obtuse triangle if any one of its angles is greater than 90 degrees

A triangle is considered an obtuse triangle if any one of its angles is greater than 90 degrees. In terms of side lengths, there are several conditions that can help to identify whether a triangle is obtuse or not.

Using the Law of Cosines, you can determine the relationship between the lengths of the sides and the angles of a triangle. The formula is given as:

c^2 = a^2 + b^2 – 2ab * cos(C)

Where:
– c is the length of the side opposite to angle C,
– a and b are the lengths of the other two sides of the triangle, and
– C is the angle opposite to side c.

To identify an obtuse triangle, you need to focus on the largest angle. Let’s assume angle C is the largest angle in the triangle.

If c^2 > a^2 + b^2, then the triangle is obtuse because the square of the longest side is greater than the sum of the squares of the other two sides.

For example, let’s say we have a triangle with side lengths of 5, 12, and 13. To determine if it is obtuse, we can use the Law of Cosines:

c^2 = 5^2 + 12^2 – 2(5)(12) * cos(C)
c^2 = 25 + 144 – 120 * cos(C)
c^2 = 169 – 120 * cos(C)

Now, we can compare this to the sum of the squares of the other two sides:

a^2 + b^2 = 5^2 + 12^2
a^2 + b^2 = 25 + 144
a^2 + b^2 = 169

In this case, we can see that c^2 > a^2 + b^2, since 169 – 120 * cos(C) is greater than 169. Therefore, this triangle is obtuse.

By applying the same method, you can determine whether any given triangle is obtuse or not, based on the relationship between the side lengths and the corresponding angles.

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