Determining If a Triangle is Obtuse: Using Side Lengths and the Pythagorean Theorem

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

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

A triangle is considered obtuse if one of its angles is greater than 90 degrees. In terms of side lengths, we can determine if a triangle is obtuse by using the Pythagorean Theorem.

The Pythagorean Theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. It can be written as:

c^2 = a^2 + b^2

In an obtuse triangle, none of the angles are equal to 90 degrees, so none of the sides are directly opposite any right angles. Therefore, we cannot use the Pythagorean Theorem to determine if a triangle is obtuse by solely looking at the side lengths.

To determine if a triangle is obtuse, we need to consider the relationship between the side lengths. In any triangle, the longest side is always opposite the largest angle, and the shortest side is opposite the smallest angle. Therefore, in an obtuse triangle, the length of the longest side is greater than the sum of the lengths of the other two sides.

Let’s say the lengths of the sides of a triangle are a, b, and c, where c is the longest side. If c > a + b, then the triangle is obtuse. If c = a + b, then the triangle is right-angled. If c < a + b, then the triangle is acute. In summary: - If c > a + b, the triangle is obtuse.
– If c = a + b, the triangle is right-angled.
– If c < a + b, the triangle is acute.

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