We want to construct a sign in the shape of the right triangle.If the longer leg of the sign is two inches more than the shorter leg and the...

We'll note the shorter leg, namely one cathetus, as
x.

The longer cathetus is x +
2.

The hypotenuse is 2x -
2.

Now, we'll apply the relation that puts together all the
sides of a right angled triangle, the Pythagorean
theorem:

hypotenuse^2 = shorter cathetus^2 + longer
cathetus^2

We'll substitute the hypotenuse and the cathetus
by the relations above:

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

We'll expand the squares from both sides and we'll
get:

4x^2 - 8x + 4 = x^2 + x^2 + 4x +
4

We'll eliminate and combine like
terms:

4x^2 - 8x = 2x^2 +
4x

We'll subtract 2x^2 + 4x both
sides:

4x^2 - 8x - 2x^2 - 4x =
0

We'll combine like
terms:

2x^2 - 12x = 0

We'll
divide by 2:

x^2 - 6x =
0

We'll factorize by x:

x(x-6)
= 0

x1 = 0

x-6 =
0

x2 = 6

Since the measure of
a cathetus cannot be zero, we'll reject the first solution and we'll keep the second
one.

So, the shorter leg of the sign has the
measure of 6 inches, the longer leg has the measure of 8 inches and the hypotenuse has
the measure of 10 inches.