\begin{tabular}{l} lie on the perpendicular bisector of \( \overline{Q R} \) with the \\ \( \begin{array}{ll}\text { 9. } Q(-5,4), R(8,-3), S(-2,-5) \\ 5)=1\end{array} \) \\ \hline\end{tabular}

We’re given two points Q(–5, 4) and R(8, –3) and a third point S(–2, –5). One common question is to determine whether S lies on the perpendicular bisector of the segment QR. (In other words, whether S is equidistant from Q and R.) We can check this in two different ways: ───────────────────────────── Method 1. Check the distances from S to Q and S to R • Find SQ:   SQ = √[ (–2 – (–5))² + (–5 – 4)² ]      = √[ (3)² + (–9)² ]      = √[ 9 + 81 ]      = √90      = 3√10 • Find SR:   SR = √[ (–2 – 8)² + (–5 – (–3))² ]      = √[ (–10)² + (–2)² ]      = √[ 100 + 4 ]      = √104      = 2√26 Since 3√10 is not equal to 2√26 (approximately 9.49 versus 10.20), S does not lie at the same distance from Q and R. Therefore, S is not on the perpendicular bisector of QR. ───────────────────────────── Method 2. Find the equation of the perpendicular bisector of QR and see if S satisfies it 1. Find the midpoint M of QR:   M = [ (–5 + 8)/2 , (4 + (–3))/2 ]     = [ 3/2 , 1/2 ] 2. Find the slope of QR:   slope of QR = (–3 – 4)/(8 – (–5)) = (–7)/13 3. The perpendicular bisector has a slope that is the negative reciprocal of –7/13:   m_perp = 13/7 4. Write the equation using point–slope form:   y – 1/2 = (13/7)(x – 3/2) 5. Test whether S(–2, –5) satisfies this equation. Substitute x = –2:   y – 1/2 = (13/7)(–2 – 3/2)       = (13/7) [ –(4/2 + 3/2) ]       = (13/7)(–7/2)       = –91/14   Then solve for y:   y = 1/2 – 91/14 = (7/14 – 91/14) = –84/14 = –6 The point calculated on the perpendicular bisector when x = –2 is (–2, –6), not (–2, –5). Hence, S does not lie on the perpendicular bisector. ───────────────────────────── Conclusion S(–2, –5) is not on the perpendicular bisector of QR because it is not equidistant from Q and R and does not satisfy the bisector’s equation.