Certainly! Let's solve this problem step-by-step:
### Understanding the Pythagorean Theorem
The Pythagorean Theorem is a fundamental relation in Euclidean geometry among the three sides of a right-angled triangle. It states that:
[tex]\[ a^2 + b^2 = c^2 \][/tex]
where:
- [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the lengths of the two legs (the shorter sides) of the right-angled triangle,
- [tex]\( c \)[/tex] is the length of the hypotenuse (the side opposite the right angle).
### Breakdown of the Options
Let's analyze each option to identify the correct one:
- Option A. the length of the longest side
- In the context of the right-angled triangle, the hypotenuse [tex]\( c \)[/tex] is always the longest side, as it is opposite the right angle.
- Option B. the length of the legs
- This option is incorrect because the legs are represented by [tex]\( a \)[/tex] and [tex]\( b \)[/tex], not [tex]\( c \)[/tex].
- Option C. the area of the triangle
- The area of a right-angled triangle is given by [tex]\(\frac{1}{2} \times a \times b\)[/tex] and not by [tex]\( c \)[/tex]. Hence, this option is also incorrect.
- Option D. the height of the triangle
- This option is incorrect because height refers to a perpendicular distance from a vertex to the opposite side, not the hypotenuse [tex]\( c \)[/tex].
### Conclusion
From our analysis, we can conclude that the hypotenuse [tex]\( c \)[/tex] represents the length of the longest side of a right-angled triangle. Therefore, the correct answer is:
A. the length of the longest side.
