Let's solve for [tex]\( a \)[/tex] using the Pythagorean Theorem, which states that for a right triangle, the square of the length of the hypotenuse ([tex]\( c \)[/tex]) is equal to the sum of the squares of the lengths of the other two sides ([tex]\( a \)[/tex] and [tex]\( b \)[/tex]):
[tex]\[ a^2 + b^2 = c^2 \][/tex]
Given values:
- [tex]\( b = 3 \)[/tex]
- [tex]\( c = 5 \)[/tex]
First, isolate [tex]\( a^2 \)[/tex] by subtracting [tex]\( b^2 \)[/tex] from both sides of the equation:
[tex]\[ a^2 = c^2 - b^2 \][/tex]
Substitute the given values into this equation:
[tex]\[ a^2 = 5^2 - 3^2 \][/tex]
Calculate the squares of [tex]\( c \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ 5^2 = 25 \][/tex]
[tex]\[ 3^2 = 9 \][/tex]
Now, subtract these values:
[tex]\[ a^2 = 25 - 9 = 16 \][/tex]
Next, solve for [tex]\( a \)[/tex] by taking the square root of both sides:
[tex]\[ a = \sqrt{16} \][/tex]
Calculate the square root:
[tex]\[ a = 4 \][/tex]
Therefore:
[tex]\[ a = 4 \][/tex]
And verifying the calculations, we find:
[tex]\[ a^2 = 16 \][/tex]
[tex]\[ a = \sqrt{16} = 4 \][/tex]
So, the value of [tex]\( a \)[/tex] is:
[tex]\[ 4 \][/tex]
