Question

pq is the perpendicular bisector of \\(\overline{st}\\). find the values of m and n.

tell whether a segment lies on a perpendicular bisector, angle bisector, or neither. justify your answer.

1. \\(\overline{ae}\\)
2. \\(\overline{dg}\\)
3. \\(\overline{fr}\\)

multiple choice.

1. if \\(\overline{jk}\\) is perpendicular to \\(\overline{xy}\\) at its midpoint m, which statement is true? (draw a picture!!!)

a. \\(jx = ky\\) b. \\(jx = kx\\) c. \\(jm = km\\) d. \\(jx\\)

1. what information is needed to conclude that \\(\overline{ef}\\) is the bisector of \\(\angle deg\\)? (draw a picture!!!)

a. \\(m\angle def = m\angle deg\\) b. \\(m\angle feg = m\angle def\\)
c. \\(m\angle ged = m\angle gef\\) d. \\(m\angle def = m\angle efg\\)

Explanation:

Step1: Solve for \( m \)

Since \( PQ \) is the perpendicular bisector of \( ST \), \( PS = PT \). So we set up the equation:
\( 3m + 9 = 5m - 13 \)
Subtract \( 3m \) from both sides:
\( 9 = 2m - 13 \)
Add 13 to both sides:
\( 22 = 2m \)
Divide by 2:
\( m = 11 \)

Step2: Solve for \( n \)

Since \( PQ \) is the perpendicular bisector of \( ST \), \( QS = QT \). So we set up the equation:
\( 6n - 3 = 4n + 14 \)
Subtract \( 4n \) from both sides:
\( 2n - 3 = 14 \)
Add 3 to both sides:
\( 2n = 17 \)
Wait, that can't be right. Wait, maybe \( QS = QT \)? Wait, looking at the diagram, \( QS \) is \( 6n - 3 \) and \( QT \) is \( 4n + 14 \)? Wait, no, maybe \( SQ = TQ \)? Wait, no, the sides from \( Q \) to \( S \) and \( Q \) to \( T \). Wait, maybe I mixed up. Wait, \( PQ \) is the perpendicular bisector, so \( PS = PT \) and \( QS = QT \). Wait, let's check again. The lengths from \( P \) to \( S \) is \( 3m + 9 \), from \( P \) to \( T \) is \( 5m - 13 \). So \( 3m + 9 = 5m - 13 \), which we solved to \( m = 11 \). Now for \( n \), the lengths from \( Q \) to \( S \) is \( 6n - 3 \), from \( Q \) to \( T \) is \( 4n + 14 \). So \( 6n - 3 = 4n + 14 \). Subtract \( 4n \): \( 2n - 3 = 14 \), add 3: \( 2n = 17 \), \( n = 8.5 \)? Wait, that seems odd. Wait, maybe \( QS = QT \) is wrong. Wait, maybe \( SQ = TQ \) is not the case. Wait, maybe \( PS = PT \) and \( QS = QT \), but maybe I made a mistake. Wait, let's check the diagram again. The quadrilateral is a kite? Wait, \( PQ \) is the perpendicular bisector, so \( S \) and \( T \) are equidistant from \( P \) and \( Q \). So \( PS = PT \) and \( QS = QT \). So for \( n \), \( 6n - 3 = 4n + 14 \). Wait, 6n -3 = 4n +14. 6n -4n =14 +3. 2n=17. n=8.5. But maybe the problem has a typo, or I misread. Wait, maybe \( QS = QT \) is \( 6n -3 = 4n +14 \), so n=8.5. But let's proceed.

Wait, maybe I made a mistake in the first step. Wait, \( 3m +9 = 5m -13 \). 5m -3m =9 +13. 2m=22. m=11. That's correct.

For \( n \), \( 6n -3 = 4n +14 \). 6n -4n =14 +3. 2n=17. n=8.5. So \( m = 11 \), \( n = 8.5 \) or \( \frac{17}{2} \).

Answer:

\( m = 11 \)
\( n = \frac{17}{2} \) (or 8.5)