Question

in quadrilateral abcd, \\( \overline { ad } \parallel \overline { bc } \\).
what must the length of segment ad be for the quadrilateral to be a
parallelogram?

Explanation:

Step1: Use the property of parallelogram

In a parallelogram, opposite sides are equal. Since \(AD\parallel BC\) and for \(ABCD\) to be a parallelogram, \(AD = BC\). So, \(3x + 7=5x - 9\).

Step2: Solve the equation for \(x\)

Subtract \(3x\) from both sides: \(3x+7 - 3x=5x - 9-3x\), which gives \(7 = 2x-9\).
Add \(9\) to both sides: \(7 + 9=2x-9 + 9\), so \(16 = 2x\).
Divide both sides by \(2\): \(x=\frac{16}{2}=8\).

Step3: Find the length of \(AD\)

Substitute \(x = 8\) into the expression for \(AD\) (\(AD=3x + 7\)).
\(AD=3\times8+7=24 + 7=31\)

Answer:

31 units