Question

the endpoints of segment (overline{mn}) have coordinates ((0,0)) and ((5,1)). the endpoints of segment (overline{ab}) have coordinates ((1\frac{1}{2},\frac{1}{4})) and ((-\frac{1}{2},k)). for what value of (k) are these segments perpendicular? answer: (k=)

Explanation:

Step1: Calculate the slope of segment \( \overline{MN} \)

The slope formula is \( m=\frac{y_2 - y_1}{x_2 - x_1} \). For points \( M(0,0) \) and \( N(5,1) \), \( m_{MN}=\frac{1 - 0}{5 - 0}=\frac{1}{5} \).

Step2: Calculate the slope of segment \( \overline{AB} \)

For points \( A(1\frac{1}{4},2\frac{1}{2})=( \frac{5}{4},\frac{5}{2}) \) and \( B(-\frac{2}{4},k)=(-\frac{1}{2},k) \), \( m_{AB}=\frac{k-\frac{5}{2}}{-\frac{1}{2}-\frac{5}{4}}=\frac{k-\frac{5}{2}}{-\frac{2 + 5}{4}}=\frac{k-\frac{5}{2}}{-\frac{7}{4}}=\frac{4(k-\frac{5}{2})}{-7}=\frac{4k - 10}{-7}\).

Step3: Use the perpendicular - slope relationship

If two lines are perpendicular, \( m_1\times m_2=-1 \). So \( m_{MN}\times m_{AB}=-1 \). Substitute \( m_{MN}=\frac{1}{5} \) and \( m_{AB}=\frac{4k - 10}{-7} \) into the equation: \(\frac{1}{5}\times\frac{4k - 10}{-7}=-1\).
Multiply both sides by \( - 35\) (the least common multiple of \(5\) and \(7\)): \(4k-10 = 35\).
Add \(10\) to both sides: \(4k=35 + 10=45\).
Divide both sides by \(4\): \(k=\frac{45}{4}=11.25\).

Answer:

\(k = \frac{45}{4}\)