Question

1. if v is the incenter of \\( \triangle pqr \\), \\( qt = 5 \\), \\( vu = 7 \\), and \\( pv = 29 \\), find each measure. \\( sv = \underline{\quad\quad} \\) \\( qs = \underline{\quad\quad} \\) \\( qv = \underline{\quad\quad} \\) \\( ps = \underline{\quad\quad} \\)

Explanation:

Step1: Identify incenter property

The incenter \( V \) is equidistant from all sides of \( \triangle PQR \), so \( SV = QT = VU = 7 \).

Step2: Calculate \( QV \) via Pythagoras

In right \( \triangle QSV \), use \( QV = \sqrt{QS^2 + SV^2} \). First, \( QS = QT = 5 \) (tangents from \( Q \) to incircle are equal).
\[
QV = \sqrt{5^2 + 7^2} = \sqrt{25 + 49} = \sqrt{74}
\]

Step3: Calculate \( PS \) via Pythagoras

In right \( \triangle PSV \), \( PV = 29 \), \( SV = 7 \).
\[
PS = \sqrt{PV^2 - SV^2} = \sqrt{29^2 - 7^2} = \sqrt{841 - 49} = \sqrt{792} = 6\sqrt{22}
\]

Answer:

\( SV = 7 \)
\( QS = 5 \)
\( QV = \sqrt{74} \)
\( PS = 6\sqrt{22} \)