Question

LMNP is a rectangle. find the value of x and the length of each diagonal.
LN = 7x + 6
MP = 9x - 2

Answer 1

To find the value of x, we need to set the lengths of LN and MP equal to each other:

7x + 6 = 9x - 2
Add 2 to both sides:
7x + 8 = 9x
Subtract 7x from both sides:
8 = 2x
Divide by 2:
x = 4

Now that we have found the value of x, we can substitute it back into LN and MP to find their lengths:

LN = 7(4) + 6
LN = 28 + 6
LN = 34

MP = 9(4) - 2
MP = 36 - 2
MP = 34

So, the length of LN and MP are 34 units each and x = 4.

To find the length of the diagonals, we can use the Pythagorean theorem since LMNP is a rectangle:

Diagonal length = √(LN^2 + MP^2)
Diagonal length = √(34^2 + 34^2)
Diagonal length = √(1156 + 1156)
Diagonal length = √(2312)
Diagonal length ≈ 48.08 units

Therefore, the length of each diagonal in the rectangle LMNP is approximately 48.08 units.