The perimeter of a triangle is 24 inches. The longest side of 4 inches is longer than the shortest side, and the shortest side is three-fourths the length of the middle side. How do you find the length of each side of the triangle?

Well this problem is simply impossible.

If the longest side is 4 inches, there is no way that the perimeter of a triangle can be 24 inches.

You are saying that 4 + (something less than 4) + (something less than 4) = 24, which is impossible.

I'd suggest that the question should read 'The perimeter of a triangle is 24 inches. The longest side is 4 inches longer than the shortest side, and the shortest side is three-fourths the length of the middle side. How do you find the length of each side of the triangle?'

Let's denote the lengths of the sides of the triangle as follows:

( x ) = length of the shortest side
( y ) = length of the middle side
( z ) = length of the longest side

Given that the longest side is 4 inches longer than the shortest side, we have ( z = x + 4 ).

Also, the shortest side is three-fourths the length of the middle side, so ( x = \frac{3}{4}y ).

The perimeter of the triangle is the sum of the lengths of its sides, which is ( x + y + z ). Since the perimeter is given as 24 inches, we have ( x + y + z = 24 ).

Substituting the expressions for ( z ) and ( x ) into the perimeter equation, we get:
( \frac{3}{4}y + y + (x + 4) = 24 )

Now, we can solve this equation for ( y ):
( \frac{7}{4}y + x + 4 = 24 )
( \frac{7}{4}y + \frac{3}{4}y + 4 = 24 ) (since ( x = \frac{3}{4}y ))
( \frac{10}{4}y + 4 = 24 )
( \frac{10}{4}y = 20 )
( y = 8 )

Now that we have found the length of the middle side (( y = 8 ) inches), we can find the lengths of the other sides using the relationships we established earlier:

For the shortest side:
( x = \frac{3}{4}y = \frac{3}{4} \times 8 = 6 ) inches

For the longest side:
( z = x + 4 = 6 + 4 = 10 ) inches

So, the lengths of the sides of the triangle are:
Shortest side: ( 6 ) inches
Middle side: ( 8 ) inches
Longest side: ( 10 ) inches