Final answer:
By recognizing that diagonals in a rectangle are congruent and setting the expressions for the diagonals equal to each other, we find that the value of x is 5. Substitute x back into any of the diagonal expressions to find that both diagonals JL and KM have lengths of 19 units.
Step-by-step explanation:
To find the lengths of the diagonals of rectangle JKLM given that JL = 3x + 4 and KM = 4x - 1, we first have to recognize that in a rectangle, the diagonals are congruent (equal in length). Therefore, the length of one diagonal JL is equal to the length of diagonal KM. To find their lengths, we can equate the two expressions for the diagonals: 3x + 4 = 4x - 1.
Subtracting 3x from both sides gives us 4 = x - 1. Adding 1 to both sides gives us 5 = x. Now that we have the value of x, we can substitute x back into the expressions for the diagonals to find their lengths. So, for diagonal JL, the length is 3(5) + 4 = 15 + 4 = 19 units. And since diagonals are congruent in a rectangle, the length of diagonal KM is also 19 units.