For what value of x must ABCD be a parallelogram?

Question

Quadrilateral ABCD has diagonals AC and BD intersecting, with the two segments formed on BD congruent, and the two segments on AC measuring 3x and 4x minus 5.​​​​​​​

(1 point)
For ABCD to be a​ parallelogram, the value of x must be

Answer 1

In a parallelogram, the diagonals bisect each other. Therefore, if the two segments formed on diagonal BD are congruent, we can write:

\[ AD + BC = 2 \cdot BD \]

Given the diagonals AC create two segments measured as \(3x\) and \(4x - 5\), for ABCD to be a parallelogram, these segments must also be equal since they are halves of the diagonals formed by their intersections.

We set the two segments equal to each other:

\[ 3x = 4x - 5 \]

Now, let's solve for \(x\):

1. Subtract \(3x\) from both sides:

\[ 0 = x - 5 \]

2. Add \(5\) to both sides:

\[ x = 5 \]

Thus, the value of \(x\) must be \( \boxed{5} \) for quadrilateral ABCD to be a parallelogram.