Final answer:
The value of x in the rectangle LMNP is calculated by setting the expressions for diagonals LN and MP equal to each other, and solving for x. After finding x = 8, the diagonals' length is determined to be 41 units each, confirming that they are equal, as is true in any rectangle.
Step-by-step explanation:
To find the value of x and the length of each diagonal in rectangle LMNP, we know that diagonals of a rectangle are equal in length. Hence, LN = MP. Setting the expressions for LN and MP equal to each other gives us the equation 2x + 25 = 6x - 7. Solving for x involves the following steps:
- Subtract 2x from both sides: 25 = 4x - 7.
- Add 7 to both sides: 32 = 4x.
- Divide by 4: x = 8.
With the value of x found, we can calculate the length of the diagonals:
- LN = 2(8) + 25 = 16 + 25 = 41.
- MP = 6(8) - 7 = 48 - 7 = 41.
This confirms that both diagonals are indeed equal, and their length is 41 units.