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 [tex]\(a\)[/tex], [tex]\(b\)[/tex], and hypotenuse [tex]\(c\)[/tex], the relation [tex]\(a^2 + b^2 = c^2\)[/tex] 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 ([tex]\(c\)[/tex]).
- [tex]\(a = 10\)[/tex], [tex]\(b = 18\)[/tex], [tex]\(c = 21\)[/tex]:
- Calculate [tex]\(a^2 + b^2\)[/tex]:
[tex]\[
10^2 + 18^2 = 100 + 324 = 424
\][/tex]
- Calculate [tex]\(c^2\)[/tex]:
[tex]\[
21^2 = 441
\][/tex]
- Check if [tex]\(424 = 441\)[/tex]: No, they are not equal.
2. Assume 18 is the hypotenuse ([tex]\(c\)[/tex]).
- [tex]\(a = 10\)[/tex], [tex]\(b = 21\)[/tex], [tex]\(c = 18\)[/tex]:
- Calculate [tex]\(a^2 + b^2\)[/tex]:
[tex]\[
10^2 + 21^2 = 100 + 441 = 541
\][/tex]
- Calculate [tex]\(c^2\)[/tex]:
[tex]\[
18^2 = 324
\][/tex]
- Check if [tex]\(541 = 324\)[/tex]: No, they are not equal.
3. Assume 10 is the hypotenuse ([tex]\(c\)[/tex]).
- [tex]\(a = 18\)[/tex], [tex]\(b = 21\)[/tex], [tex]\(c = 10\)[/tex]:
- Calculate [tex]\(a^2 + b^2\)[/tex]:
[tex]\[
18^2 + 21^2 = 324 + 441 = 765
\][/tex]
- Calculate [tex]\(c^2\)[/tex]:
[tex]\[
10^2 = 100
\][/tex]
- Check if [tex]\(765 = 100\)[/tex]: 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
