Final answer:
To solve for x, set the expressions for diagonals LN and MP equal to one another, giving 5x + 30 = 9x - 6. Solving this equation finds x equals 9. Both diagonals are 75 units long since LN and MP are equal in a rectangle.
Step-by-step explanation:
The student is asked to find the value of x and the length of each diagonal in a rectangle LMNP, given that LN = 5x + 30 and MP = 9x - 6.
Finding the Value of x:
In a rectangle, the opposite sides are equal in length, so side LN equals side MP. Setting the expressions for LN and MP equal to each other gives us an equation to solve for x:
5x + 30 = 9x - 6
4x = 36
x = 9
Length of the Diagonals:
Since LN is equal to MP in a rectangle and both represent the length of the diagonals, we can substitute x into either LN or MP to find the diagonal length:
LN = 5(9) + 30 = 45 + 30 = 75
MP = 9(9) - 6 = 81 - 6 = 75
Therefore, both diagonals LN and MP are 75 units long.