The perimeter of the triangle is 30 units.
How to form equation from word problem.
From the description
The triangle is a right-angled triangle
let hypotenuse = x
longer side = x - 1
shorter leg = x-8
Using Pythagorean theorem
hyp² = longer leg² + shorter leg²
x² = (x-1)² + (x-8)²
Expand
x² = (x-1)(x-1)+(x-8)(x-8)
x² = (x²-x-x + 1)+(x²-8x-8x +64)
x² = (x² -2x + 1) + (x² - 16x + 64)
Collect like terms
x² - x² -x² +2x +16x -1 -64 = 0
-x² + 18x -65 =0
x² -18x +65 = 0
let's solve the quadratic equation
x² - 18x + 65 = 0 .
We can factor it as (x-13)(x-5) = 0.
So, the solutions are x = 13 and x = 5.
x cannot be 5 because x -8: 5-8 = -3
hypotenuse , x = 13
Longer leg, x-1 = 13 -1 = 12
Shorter leg, x-8 = 13-8 = 5
Perimeter = hypotenuse+longer leg + shorter leg
P = 13 + 12 + 5
= 30 units