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
Answers
GPT-4o mini
In a parallelogram, the diagonals bisect each other. This means that the segments formed by the intersection point of the diagonals (in this case, point P) are equal in length.
Since \( P \) is the midpoint of diagonals \( LN \) and \( MO \), we can express this relationship mathematically using the lengths given:
\[
LP = PN
\]
According to the problem, we have:
\[
LP = 3x - 4
\]
\[
PN = x + 14
\]
Setting these two expressions equal to each other, we get:
\[
3x - 4 = x + 14
\]
Now, 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 \( 9 \).
Since \( P \) is the midpoint of diagonals \( LN \) and \( MO \), we can express this relationship mathematically using the lengths given:
\[
LP = PN
\]
According to the problem, we have:
\[
LP = 3x - 4
\]
\[
PN = x + 14
\]
Setting these two expressions equal to each other, we get:
\[
3x - 4 = x + 14
\]
Now, 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 \( 9 \).