The longer leg of a triangle is one foot less than twice the shorter leg. The hypotenuse is 17 feet long. Find the area of the triangle.

short leg --- x

longer leg --- 2x-1

x^2 + (2x-1)^2 = 17^2
x^2 + 4x^2 - 4x + 1 = 289
5x^2 - 4x - 288 = 0
(x-8)(5x + 36) = 0
x = 8 or x is an inadmissable negative number

short leg = 8 ft
long leg = 15 ft

area = (1/2)(8)(15) = 60 ft^2

To find the area of a triangle, we need the lengths of its base and height. In this case, the base and height refer to the legs of the right triangle.

Let's assume the shorter leg of the triangle is represented by the variable "x."

According to the problem, the longer leg is one foot less than twice the shorter leg. So, the longer leg can be represented as "2x - 1."

We also know that the hypotenuse of the triangle is 17 feet.

Using the Pythagorean theorem, we can form the equation: shorter leg² + longer leg² = hypotenuse².

Substituting the values into the equation, we get:

x² + (2x - 1)² = 17²

Expanding the equation:

x² + 4x² - 4x + 1 = 289

Combining like terms:

5x² - 4x + 1 - 289 = 0

Simplifying the equation:

5x² - 4x - 288 = 0

To solve this quadratic equation, we can use the quadratic formula:

x = [ -b ± √(b² - 4ac) ] / 2a

In our equation, a = 5, b = -4, and c = -288.

Substituting the values into the quadratic formula:

x = [ -(-4) ± √((-4)² - 4(5)(-288)) ] / (2 *5)

Solving the equation yields two possible values for x:

x ≈ 12.64 or x ≈ -9.04

Since we are dealing with lengths, the value of x cannot be negative. Thus, x ≈ 12.64 feet.

Now that we have the value of the shorter leg, we can find the longer leg:

Longer leg = 2x - 1
Longer leg = 2(12.64) - 1
Longer leg ≈ 25.28 - 1
Longer leg ≈ 24.28 feet

Now that we have both the shorter leg (12.64 feet) and the longer leg (24.28 feet), we can calculate the area of the triangle:

Area = (1/2) * base * height
Area = (1/2) * 12.64 * 24.28
Area ≈ 153.516 square feet

Therefore, the area of the triangle is approximately 153.516 square feet.