To determine the length of the remaining side of a right triangle when we know the lengths of the hypotenuse and one of the other sides, we can use the Pythagorean theorem. The Pythagorean theorem states:
[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.
Given:
- The length of the hypotenuse [tex]\( c \)[/tex] is 50 meters.
- The length of one side [tex]\( a \)[/tex] is 30 meters.
We need to find the length of the remaining side [tex]\( b \)[/tex]. According to the Pythagorean theorem, we can rearrange the formula to solve for [tex]\( b \)[/tex]:
[tex]\[ b^2 = c^2 - a^2 \][/tex]
Let's substitute the known values into this formula:
[tex]\[ b^2 = 50^2 - 30^2 \][/tex]
First, calculate [tex]\( 50^2 \)[/tex]:
[tex]\[ 50^2 = 2500 \][/tex]
Next, calculate [tex]\( 30^2 \)[/tex]:
[tex]\[ 30^2 = 900 \][/tex]
Now subtract [tex]\( 30^2 \)[/tex] from [tex]\( 50^2 \)[/tex]:
[tex]\[ b^2 = 2500 - 900 \][/tex]
[tex]\[ b^2 = 1600 \][/tex]
To find [tex]\( b \)[/tex], we take the square root of both sides of the equation:
[tex]\[ b = \sqrt{1600} \][/tex]
Finally, calculate the square root of 1600:
[tex]\[ b = 40 \][/tex]
Thus, the length of the remaining side is 40 meters.
