To find the length of [tex]\(c\)[/tex] in a right triangle where the legs [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are given as 5 and 6, respectively, follow these steps:
1. Identify the sides of the right triangle:
- One leg of the right triangle ([tex]\(a\)[/tex]) is 5.
- The other leg ([tex]\(b\)[/tex]) is 6.
- The hypotenuse ([tex]\(c\)[/tex]) is the side we need to find.
2. Recall the Pythagorean theorem:
[tex]\[
c^2 = a^2 + b^2
\][/tex]
This theorem relates the lengths of the sides in a right triangle: the square of the hypotenuse is equal to the sum of the squares of the other two sides.
3. Substitute the known values [tex]\(a\)[/tex] and [tex]\(b\)[/tex] into the theorem:
[tex]\[
c^2 = 5^2 + 6^2
\][/tex]
4. Calculate the squares of [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
[tex]\[
5^2 = 25
\][/tex]
[tex]\[
6^2 = 36
\][/tex]
5. Add these squares together:
[tex]\[
c^2 = 25 + 36
\][/tex]
[tex]\[
c^2 = 61
\][/tex]
6. Take the square root of both sides to solve for [tex]\(c\)[/tex]:
[tex]\[
c = \sqrt{61}
\][/tex]
7. By taking the square root, we find the length of the hypotenuse:
[tex]\[
c \approx 7.810249675906654
\][/tex]
So, the length of [tex]\(c\)[/tex] is approximately 7.810.
