To determine whether a triangle with sides of lengths 10, 18, and 21 is a right triangle, we need to verify that the triangle satisfies the Pythagorean theorem. The Pythagorean theorem states that in a right triangle with sides \(\(a\)\), \(\(b\)\), and hypotenuse \(\(c\)\), the relation \(\(a^2 + b^2 = c^2\)\) must hold true.
Let's check all possibilities to see if any combination of these sides satisfies the Pythagorean theorem.
1. Assume 21 is the hypotenuse (\(\(c\)\)).
- \(\(a = 10\)\), \(\(b = 18\)\), \(\(c = 21\)\):
- Calculate \(\(a^2 + b^2\)\):
\(\[
10^2 + 18^2 = 100 + 324 = 424
\]\)
- Calculate \(\(c^2\)\):
\(\[
21^2 = 441
\]\)
- Check if \(\(424 = 441\)\): No, they are not equal.
2. Assume 18 is the hypotenuse (\(\(c\)\)).
- \(\(a = 10\)\), \(\(b = 21\)\), \(\(c = 18\)\):
- Calculate \(\(a^2 + b^2\)\):
\(\[
10^2 + 21^2 = 100 + 441 = 541
\]\)
- Calculate \(\(c^2\)\):
\(\[
18^2 = 324
\]\)
- Check if \(\(541 = 324\)\): No, they are not equal.
3. Assume 10 is the hypotenuse (\(\(c\)\)).
- \(\(a = 18\)\), \(\(b = 21\)\), \(\(c = 10\)\):
- Calculate \(\(a^2 + b^2\)\):
\(\[
18^2 + 21^2 = 324 + 441 = 765
\]\)
- Calculate \(\(c^2\)\):
\(\[
10^2 = 100
\]\)
- Check if \(\(765 = 100\)\): No, they are not equal.
Since none of the combinations satisfy the Pythagorean theorem, the triangle with sides 10, 18, and 21 is not a right triangle.
Therefore, the correct answer is:
B. False
