Determining if a Triangle is Acute Based on Side Lengths and the Triangle Inequality Theorem

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

Regarding the side lengths of a triangle, a triangle is considered an acute triangle if all three of its angles are acute angles

Regarding the side lengths of a triangle, a triangle is considered an acute triangle if all three of its angles are acute angles. In other words, all angles of an acute triangle measure less than 90 degrees.

To determine if a triangle is acute based on its side lengths, we need to apply the Triangle Inequality Theorem. According to this theorem, the sum of the lengths of any two sides of a triangle must always be greater than the length of the third side.

Let’s say we have a triangle with side lengths a, b, and c. To determine if it is an acute triangle, we need to check if a^2 + b^2 > c^2, b^2 + c^2 > a^2, and c^2 + a^2 > b^2. If all three of these inequalities are true, then the triangle is acute.

For example, let’s consider a triangle with side lengths 5, 7, and 9. We can check if it is an acute triangle by applying the inequalities:

5^2 + 7^2 > 9^2
25 + 49 > 81
74 > 81

7^2 + 9^2 > 5^2
49 + 81 > 25
130 > 25

9^2 + 5^2 > 7^2
81 + 25 > 49
106 > 49

Since all three inequalities are true, the triangle with side lengths 5, 7, and 9 is an acute triangle.

Remember, for a triangle to be acute, all three angles must be acute. If any of the angles is obtuse (greater than 90 degrees) or right (equal to 90 degrees), then the triangle is not acute.

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