To find the length of the hypotenuse in a right triangle, we can use the Pythagorean Theorem. The theorem states that in 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]. Mathematically, this is expressed as:
[tex]\[ c^2 = a^2 + b^2 \][/tex]
Given that [tex]\(a = 6\)[/tex] and [tex]\(b = 9\)[/tex], we need to find [tex]\(c\)[/tex].
1. First, square the lengths of [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
[tex]\[ a^2 = 6^2 = 36 \][/tex]
[tex]\[ b^2 = 9^2 = 81 \][/tex]
2. Next, add these squares together to find the value of [tex]\(c^2\)[/tex]:
[tex]\[ c^2 = 36 + 81 = 117 \][/tex]
3. To find [tex]\(c\)[/tex], take the square root of [tex]\(c^2\)[/tex]:
[tex]\[ c = \sqrt{117} \][/tex]
4. Calculating the square root of 117 gives us approximately:
[tex]\[ c \approx 10.816653826391969 \][/tex]
5. Finally, round this value to three decimal places:
[tex]\[ c \approx 10.817 \][/tex]
So, the length of the hypotenuse, rounded to three decimal places, is [tex]\(10.817\)[/tex].
