To find the length of the missing side [tex]\( b \)[/tex] in a right triangle where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the legs and [tex]\( c \)[/tex] is the hypotenuse, we use the Pythagorean theorem:
[tex]\[
a^2 + b^2 = c^2
\][/tex]
Given:
[tex]\[
a = 9, \quad c = 13
\][/tex]
We need to find [tex]\( b \)[/tex]. According to the Pythagorean theorem, we can solve for [tex]\( b^2 \)[/tex]:
[tex]\[
b^2 = c^2 - a^2
\][/tex]
Substitute the values of [tex]\( a \)[/tex] and [tex]\( c \)[/tex] into the equation:
[tex]\[
b^2 = 13^2 - 9^2
\][/tex]
Calculate [tex]\( 13^2 \)[/tex] and [tex]\( 9^2 \)[/tex]:
[tex]\[
13^2 = 169
\][/tex]
[tex]\[
9^2 = 81
\][/tex]
Now, subtract [tex]\( 9^2 \)[/tex] from [tex]\( 13^2 \)[/tex]:
[tex]\[
b^2 = 169 - 81 = 88
\][/tex]
We have determined that [tex]\( b^2 = 88 \)[/tex]. To find [tex]\( b \)[/tex], we need to take the square root of [tex]\( 88 \)[/tex]:
[tex]\[
b = \sqrt{88}
\][/tex]
Simplifying [tex]\( \sqrt{88} \)[/tex]:
[tex]\[
\sqrt{88} = \sqrt{4 \times 22} = \sqrt{4} \times \sqrt{22} = 2\sqrt{22}
\][/tex]
Therefore, the length of side [tex]\( b \)[/tex] is:
[tex]\[
b = 2\sqrt{22}
\][/tex]
So the correct answer is:
[tex]\[
2 \sqrt{22}
\][/tex]
