How to Determine if a Triangle is Acute: Using Pythagorean Theorem and Triangle Inequality

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

A triangle is considered an acute triangle if all three of its angles are less than 90 degrees

A triangle is considered an acute triangle if all three of its angles are less than 90 degrees. In terms of side lengths, we can use the Pythagorean Theorem and the Triangle Inequality Theorem to determine if a triangle is acute.

The Pythagorean Theorem states that in a right triangle, the sum of the squares of the two legs (sides adjacent to the right angle) is equal to the square of the hypotenuse (side opposite the right angle). It can be represented as:

a^2 + b^2 = c^2

where ‘a’ and ‘b’ are the leg lengths and ‘c’ is the hypotenuse length.

For an acute triangle, none of the angles are 90 degrees. As a result, none of the sides will be the hypotenuse. So, if we consider the largest side of an acute triangle, it will be smaller than the square root of the sum of squares of the other two sides.

This can be represented mathematically as:

c < √(a^2 + b^2) Using the Triangle Inequality Theorem, we can further determine if a triangle is acute by comparing the sum of the lengths of any two sides to the length of the third side. According to the Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Mathematically: a + b > c
a + c > b
b + c > a

If all three inequalities above are satisfied, then the triangle is acute.

In summary, for a triangle to be an acute triangle:

1. All three angles must be less than 90 degrees.
2. The largest side length of the triangle must be smaller than the square root of the sum of squares of the other two side lengths.
3. The sum of the lengths of any two sides must be greater than the length of the third side.

More Answers: