To determine the length of the hypotenuse of a right triangle whose legs measure [tex]\(3x\)[/tex] and [tex]\(4x\)[/tex] (with [tex]\(x > 0\)[/tex]), we can employ the Pythagorean theorem. The Pythagorean 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]). The formula is:
[tex]\[
c^2 = a^2 + b^2
\][/tex]
Here, [tex]\(a = 3x\)[/tex] and [tex]\(b = 4x\)[/tex]. Let's follow these steps to find the hypotenuse [tex]\(c\)[/tex]:
1. Square the lengths of the legs:
[tex]\[
(3x)^2 = 9x^2
\][/tex]
[tex]\[
(4x)^2 = 16x^2
\][/tex]
2. Sum the squares of the legs:
[tex]\[
9x^2 + 16x^2 = 25x^2
\][/tex]
3. Take the square root of the sum to find the hypotenuse:
[tex]\[
c = \sqrt{25x^2}
\][/tex]
4. Simplify the square root:
[tex]\[
c = \sqrt{25} \cdot \sqrt{x^2}
\][/tex]
[tex]\[
c = 5 \cdot |x|
\][/tex]
Since [tex]\(x > 0\)[/tex], [tex]\(|x| = x\)[/tex], so the hypotenuse [tex]\(c\)[/tex] simplifies to:
[tex]\[
c = 5x
\][/tex]
Therefore, the length of the hypotenuse of the right triangle is given by [tex]\(5x\)[/tex]. The correct answer is:
C) [tex]\(5x\)[/tex]
