To solve for the missing side [tex]\( b \)[/tex] of a right triangle, given that [tex]\( a = 9 \)[/tex] (one of the legs) and [tex]\( c = 13 \)[/tex] (the hypotenuse), we will use the Pythagorean theorem:
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse [tex]\( c \)[/tex] is equal to the sum of the squares of the other two sides [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[
c^2 = a^2 + b^2
\][/tex]
Given:
[tex]\[
a = 9 \quad \text{and} \quad c = 13
\][/tex]
Step 1: Substitute the given values into the Pythagorean theorem:
[tex]\[
13^2 = 9^2 + b^2
\][/tex]
Step 2: Calculate the squares:
[tex]\[
169 = 81 + b^2
\][/tex]
Step 3: Isolate [tex]\( b^2 \)[/tex] by subtracting 81 from both sides:
[tex]\[
169 - 81 = b^2
\][/tex]
[tex]\[
88 = b^2
\][/tex]
Step 4: To find [tex]\( b \)[/tex], take the square root of both sides:
[tex]\[
b = \sqrt{88}
\][/tex]
The result [tex]\( \sqrt{88} \)[/tex] can also be simplified:
[tex]\[
b = \sqrt{4 \times 22} = 2 \sqrt{22}
\][/tex]
Therefore, the missing side [tex]\( b \)[/tex] is [tex]\( 2 \sqrt{22} \)[/tex].
So the length of the side not given is:
[tex]\[
b = 2 \sqrt{22}
\][/tex]
