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 the slope of line \(t\)

Pick two points on line \(t\), say \((- 2,-1)\) and \((2,2)\). The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). So, \(m_t=\frac{2-(-1)}{2 - (-2)}=\frac{3}{4}\).

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

Since \(\overline{AB}\perp t\), the product of their slopes is \(- 1\). Let the slope of \(\overline{AB}\) be \(m_{AB}\). Then \(m_{AB}\times m_t=-1\), so \(m_{AB}=-\frac{4}{3}\).

Step3: Calculate the slope of \(\overline{AB}\) using points \(A(4,2)\) and \(B(-1,y)\)

The slope formula for \(\overline{AB}\) is \(m_{AB}=\frac{y - 2}{-1 - 4}=\frac{y - 2}{-5}\).

Step4: Solve for \(y\)

Set \(\frac{y - 2}{-5}=-\frac{4}{3}\). Cross - multiply: \(3(y - 2)=20\). Expand: \(3y-6 = 20\). Add 6 to both sides: \(3y=26\). Divide by 3: \(y=\frac{26}{3}\).

Answer:

\(y = \frac{26}{3}\)