A sign in the shape of a right triangle has one leg that is 7 in longer than the other leg what is the length of the shorter leg if the perimeter is 30 in.

Let the shorter leg be x.

Let the hypotenuse be y.

The the longer leg is x + 7.

The perimeter is 30.

x + x + 7 + y = 30

y = 30 - 2x - 7

y = - 2x + 23

The hypotenuse is - 2x + 23.

Now we use the Pythagorean theorem.

a^2 + b^2 = c^2

x^2 + (x + 7) ^2 = (-2x + 23) ^2

x^2 + x^2 + 14x + 49 = 4x^2 - 92x + 529

2x^2 - 106x + 480 = 0

x^2 - 53x + 240 = 0

(x - 5) (x - 48) = 0

x = 5 or x = 48

Since 48 is greater than the perimeter, we discard that solution.

x is the shorter leg.

The shorter leg measures 5 units.