Question

1. given a(4, 2) and b(-1, y) and the graph of line t below, find the value of y so that $overrightarrow{ab}perp t$.

Explanation:

Step1: Find the slope of line \(t\)

Pick two points on line \(t\), say \((0, - 2)\) and \((2,0)\). The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). So, \(m_t=\frac{0-( - 2)}{2 - 0}=\frac{2}{2}=1\).

Step2: Find the slope of \(\overrightarrow{AB}\)

The slope of the line passing through \(A(4,2)\) and \(B(-1,y)\) is \(m_{AB}=\frac{y - 2}{-1 - 4}=\frac{y - 2}{-5}\).

Step3: Use the perpendicular - slope relationship

If two lines are perpendicular, the product of their slopes is \(- 1\). So, \(m_{AB}\times m_t=-1\). Substitute \(m_t = 1\) and \(m_{AB}=\frac{y - 2}{-5}\) into the equation: \(\frac{y - 2}{-5}\times1=-1\).

Step4: Solve for \(y\)

Multiply both sides of the equation \(\frac{y - 2}{-5}=-1\) by \(-5\): \(y - 2 = 5\). Then add \(2\) to both sides: \(y=5 + 2=7\).

Answer:

\(y = 7\)