Question

1. triangle qrs has a perimeter of 55 units. segment rt bisects \\( \angle r \\). what is the length of \\( \overline{qt} \\)? 4. \\( qt= \\)____

Explanation:

Step1: Find the length of \(QR + QS\)

The perimeter of \(\triangle QRS\) is \(55\), and \(RS = 18\), \(TS=12\).
Since \(QR + QS+RS + TS=55\), then \(QR + QS=55-(18 + 12)=25\).

Step2: Use the Angle - Bisector Theorem

By the Angle - Bisector Theorem, \(\frac{QT}{TS}=\frac{QR}{RS}\), let \(QT = x\), then \(QR = 25 - x\).
Substitute into the theorem: \(\frac{x}{12}=\frac{25 - x}{18}\).
Cross - multiply: \(18x=12\times(25 - x)\).
Expand: \(18x = 300-12x\).
Add \(12x\) to both sides: \(18x+12x=300\), \(30x = 300\).
Solve for \(x\): \(x = 10\).

Answer:

\(10\)