To determine whether a triangle with sides of lengths 9, 22, and 24 is a right triangle, we need to apply the Pythagorean theorem.
The Pythagorean theorem states that for a triangle to be a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) should be equal to the sum of the squares of the lengths of the other two sides. This can be expressed as:
[tex]\[ c^2 = a^2 + b^2 \][/tex]
where [tex]\( c \)[/tex] is the length of the hypotenuse and [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the lengths of the other two sides.
Firstly, identify the largest side, which is 24. This will be [tex]\( c \)[/tex]. Then, [tex]\( a \)[/tex] and [tex]\( b \)[/tex] will be 9 and 22 respectively.
Let's check this:
1. Calculate [tex]\( a^2 \)[/tex]:
[tex]\[ 9^2 = 81 \][/tex]
2. Calculate [tex]\( b^2 \)[/tex]:
[tex]\[ 22^2 = 484 \][/tex]
3. Calculate [tex]\( c^2 \)[/tex]:
[tex]\[ 24^2 = 576 \][/tex]
4. Finally, check if [tex]\( a^2 + b^2 = c^2 \)[/tex]:
[tex]\[ 81 + 484 = 565 \][/tex]
We see that:
[tex]\[ 565 \neq 576 \][/tex]
Therefore, the sum of the squares of the two shorter sides (81 + 484) does not equal the square of the longest side (576).
Hence, the triangle with sides of lengths 9, 22, and 24 is not a right triangle.
The correct answer is:
B. False
