Question

Parallelogram LMNO has two diagonal lines, LN¯¯¯¯¯¯¯¯ and MO¯¯¯¯¯¯¯¯¯ , that intersect at P . The length of LP¯¯¯¯¯¯¯ can be expressed by 3x−4 , and PN¯¯¯¯¯¯¯¯ by the expression x+14 . Identify a pattern regarding the diagonals of the parallelogram to find the value of x.(1 point)
x=

Answer 1

In a parallelogram, the diagonals bisect each other. This means that the lengths of the segments created by the intersection of the diagonals are equal.

Given that the length of \( LP \) is described by the expression \( 3x - 4 \), and the length of \( PN \) is described by \( x + 14 \), we set these two expressions equal to each other because \( P \) is the midpoint of both diagonals:

\[
LP = PN
\]

Substituting the expressions, we have:

\[
3x - 4 = x + 14
\]

Now, we solve for \( x \):

1. Subtract \( x \) from both sides:
\[
3x - x - 4 = 14
\]
\[
2x - 4 = 14
\]

2. Add 4 to both sides:
\[
2x = 18
\]

3. Divide both sides by 2:
\[
x = 9
\]

Thus, the value of \( x \) is:

\[
\boxed{9}
\]