Question

q is the midpoint of \\(\overline{ru}\\) and t is the midpoint of \\(\overline{su}\\). if \\(rs = x + 8\\) and \\(qt = x\\), what is \\(qt\\)?

Explanation:

Step1: Use the mid - segment theorem

The mid - segment theorem states that the length of the mid - segment (a segment connecting the midpoints of two sides of a triangle) is half the length of the third side. Here, \(QT\) is the mid - segment of \(\triangle RSU\) (since \(Q\) is the midpoint of \(RU\) and \(T\) is the midpoint of \(SU\)), so \(QT=\frac{1}{2}RS\).

Step2: Substitute the given expressions

We know that \(RS = x + 8\) and \(QT=x\). Substituting into the equation \(QT=\frac{1}{2}RS\), we get \(x=\frac{1}{2}(x + 8)\).

Step3: Solve the equation for \(x\)

Multiply both sides of the equation \(x=\frac{1}{2}(x + 8)\) by \(2\) to get \(2x=x + 8\).
Subtract \(x\) from both sides: \(2x-x=x + 8-x\), which simplifies to \(x = 8\).

Answer:

\(8\)