Question

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

Explanation:

Step1: Find slope of line \(t\)

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

Step2: Determine slope of \(\overline{AB}\)

If two lines are perpendicular, the product of their slopes is \(- 1\). Let the slope of \(\overline{AB}\) be \(m_{AB}\). Since \(m_t\times m_{AB}=-1\) and \(m_t = 1\), then \(m_{AB}=-1\).

Step3: Calculate \(y\) using slope formula

The slope formula for two - points \(A(4,2)\) and \(B(-1,y)\) is \(m_{AB}=\frac{y - 2}{-1 - 4}\). Since \(m_{AB}=-1\), we have \(\frac{y - 2}{-1 - 4}=-1\). Cross - multiply: \(y−2=-1\times(-5)\). Then \(y−2 = 5\). Add 2 to both sides: \(y=5 + 2=7\).

Answer:

\(y = 7\)