- Verify the triangle existence using the triangle inequality: \(2+4 > 5\).
- Calculate the square of the longest side: \(5^2 = 25\).
- Calculate the sum of the squares of the two shorter sides: \(2^2 + 4^2 = 20\).
- Compare the values: since \(20 < 25\), the triangle is obtuse.
- Conclude that the correct description is \(\boxed{\text{The triangle is not acute because } 2^2 + 4^2 < 5^2}\).
Explanation
- Verify the triangle existence.
To determine the nature of a triangle with side lengths 2, 4, and 5, we first ensure it satisfies the triangle inequality, which states that the sum of the two shorter sides must be greater than the longest side. Here, the sides are 2, 4, and 5. The two shorter sides are 2 and 4. Since \(2 + 4 = 6\), and \(6 > 5\), the triangle inequality is satisfied, and these sides indeed form a valid triangle.
- Calculate squares and compare values.
To classify the triangle as acute, right, or obtuse, we compare the square of the longest side with the sum of the squares of the other two sides.
The longest side is \(c = 5\).
The square of the longest side is \(5^2 = 25\).
The sum of the squares of the two shorter sides (a = 2 and b = 4) is \(2^2 + 4^2 = 4 + 16 = 20\).
- Classify the triangle.
According to the Pythagorean theorem generalization (the law of cosines), for a triangle with sides \(a\), \(b\), and longest side \(c\):
- If \(a^2 + b^2 = c^2\), the triangle is right.
- If \(a^2 + b^2 > c^2\), the triangle is acute.
- If \(a^2 + b^2 < c^2\), the triangle is obtuse.
Comparing our values: \(20 < 25\), which means \(2^2 + 4^2 < 5^2\). Since the sum of the squares of the two shorter sides is less than the square of the longest side, the triangle is obtuse (not acute). Therefore, the correct statement is that the triangle is not acute because \(2^2 + 4^2 < 5^2\).
Examples
Understanding how to classify triangles is essential in fields like construction and engineering. For example, a carpenter building a support frame for a roof needs to know if the joints are acute or obtuse to ensure structural stability. Using the relationship between the squares of the side lengths, \(a^2 + b^2\) compared to \(c^2\), they can quickly determine if they have created the correct angle for their design needs without needing a protractor for every single cut.