Answer:
x=3
Explanation:
In a right triangle, the Pythagorean theorem is applicable. According to this theorem, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. We can write this as:
`a² + b² = c²`
Given your triangle, we can substitute these values into the theorem:
`(3)² + (3x - 5)² = (5)²`
Solving for x, we get:
`9 + 9x² - 30x + 25 = 25`
Simplifying the equation, we get:
`9x² - 30x + 9 = 0`
Divide by 9:
`x² - (30/9)x + 1 = 0`
`x² - (10/3)x + 1 = 0`
This is a quadratic equation in the form `ax² + bx + c = 0` and its solutions are given by the quadratic formula:
`x = [-b ± sqrt(b² - 4ac)] / 2a`
Substituting a = 1, b = -10/3, and c = 1 into the formula, we have:
`x = [(10/3) ± sqrt((10/3)² - 4*1*1)] / 2*1`
`x = [(10/3) ± sqrt((100/9) - 4)] / 2`
`x = [(10/3) ± sqrt((100 - 36) / 9)] / 2`
`x = [(10/3) ± sqrt(64/9)] / 2`
`x = [(10/3) ± (8/3)] / 2`
Solving for x, we get two possible solutions:
1. `x = [(10 + 8) / 3] / 2 = 18/6 = 3`
1. `x = [(10 - 8) / 3] / 2 = 2/6 = 1/3`
However, if x = 1/3, the second leg length would be 3(1/3) - 5 = 1 - 5 = -4, which is not possible in a triangle.
So, the only valid solution is `x = 3`.