. The shorter leg of a right triangle is 8 less than the hypotenu se. The longer leg is 1 less than the hypotenu ise. Find the perimeter of the triangle

Let's denote the hypotenuse of the right triangle as "h". According to the problem, the shorter leg is "h - 8" and the longer leg is "h - 1".<br /><br />Using the Pythagorean theorem, we can write the equation:<br /><br />(h - 8)^2 + (h - 1)^2 = h^2<br /><br />Expanding and simplifying the equation:<br /><br />h^2 - 16h + 64 + h^2 - 2h + 1 = h^2<br /><br />2h^2 - 18h + 65 = h^2<br /><br />h^2 - 18h + 65 = 0<br /><br />Now, we can solve this quadratic equation to find the value of "h":<br /><br />h = [18 ± sqrt(18^2 - 4 * 65)] / 2<br /><br />h = [18 ± sqrt( 260)] / 2<br /><br />h = [18 ± sqrt(64)] / 2<br /><br />h = [18 ± 8] / 2<br /><br />So, we have two possible values for "h":<br /><br />h = (18 + 8) / 2 = 13<br /><br />h = (18 - 8) / 2 = 5<br /><br />Since the hypotenuse of a triangle must be the longest side, we discard the value h = 5.<br /><br />Therefore, the hypotenuse of the triangle is 13 units. The shorter leg is 13 - 8 = 5 units, and the longer leg is 13 - 1 = 12 units.<br /><br />The perimeter of the triangle is the sum of all its sides:<br /><br />Perimeter = 13 + 5 + 12 = 30 units.