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Parallelogram Proofs Practice

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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=

1 answer

In a parallelogram, the diagonals bisect each other. This means that the lengths of segments from the intersection point (P) to the vertices on each diagonal are equal.

In this case, you have:

  • \( LP = 3x - 4 \)
  • \( PN = x + 14 \)

Since the diagonals bisect each other, we can set the two expressions equal to each other:

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

Now, solve for \( x \):

  1. Subtract \( x \) from both sides:

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

  1. Simplify:

\[ 2x - 4 = 14 \]

  1. Add 4 to both sides:

\[ 2x = 18 \]

  1. Divide by 2:

\[ x = 9 \]

So, the value of \( x \) is \( 9 \).