Final answer:
To determine if a triangle with sides of 27, 17, and 13 is right, acute, or obtuse, the Pythagorean Theorem is applied. The results, showing that 17² + 13² is less than 27², indicate the triangle is obtuse since the angle opposite the longest side is greater than 90 degrees.
Explanation:
To determine if a triangle with sides of 27, 17, and 13 is right, acute, or obtuse, we use the Pythagorean Theorem which 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. This can be written as a² + b² = c², where c represents the length of the hypotenuse, and a and b represent the lengths of the other two sides.
For this triangle, we assume the longest side, 27, to be the hypotenuse. We then calculate 17² + 13² and compare it to 27² to see if they are equal. After calculating, we get:
- 17² = 289
- 13² = 169
- 289 + 169 = 458
- 27² = 729
Since 458 is less than 729, the triangle is not right-angled. Instead, it is an obtuse triangle because the sum of the squares of the two shorter sides is less than the square of the longest side, indicating that the angle opposite the longest side is greater than 90 degrees.