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 \( t \)

From the graph, line \( t \) passes through \((-4, 0)\) and \((2, 3)\) (estimating grid points). Slope \( m_t = \frac{3 - 0}{2 - (-4)} = \frac{3}{6} = \frac{1}{2} \).

Step2: Slope of perpendicular line

If two lines are perpendicular, their slopes multiply to \(-1\). Let slope of \( \overline{AB} \) be \( m_{AB} \). Then \( m_{AB} \times \frac{1}{2} = -1 \), so \( m_{AB} = -2 \).

Step3: Calculate \( y \) using slope formula

Slope between \( A(4, 2) \) and \( B(-1, y) \) is \( m_{AB} = \frac{y - 2}{-1 - 4} = \frac{y - 2}{-5} \). Set equal to \(-2\): \( \frac{y - 2}{-5} = -2 \). Multiply both sides by \(-5\): \( y - 2 = 10 \). Add 2: \( y = 12 \). Wait, correction: Wait, maybe better grid points. Let's recheck line \( t \). Let's take two clear points: left point \((-4, 0)\) and right point \((2, 3)\)? Wait, maybe another pair. Wait, maybe the line passes through \((-3, -1)\) and \((3, 2)\)? Wait, no, let's look at the grid. The left dot is at \((-4, 0)\) (since x=-4, y=0), and the right dot is at (2, 3)? Wait, no, maybe ( -4, 0 ) and ( 2, 3 ) gives slope 3/6=1/2. Then perpendicular slope is -2. Then slope of AB: (y - 2)/(-1 - 4) = (y - 2)/(-5) = -2. So y - 2 = 10? Wait, no: (y - 2)/(-5) = -2 → y - 2 = (-2)(-5) = 10 → y = 12? Wait, that seems high. Wait, maybe I picked wrong points. Let's check again. Wait, maybe the line t has slope 1/2? Wait, maybe the two points are (-4, 0) and (0, 2)? Wait, from (-4,0) to (0,2): slope (2-0)/(0 - (-4))=2/4=1/2. Yes, that's better. So line t passes through (-4, 0) and (0, 2), so slope 1/2. Then perpendicular slope is -2. Then slope of AB: (y - 2)/(-1 - 4) = (y - 2)/(-5) = -2. So y - 2 = (-2)(-5)=10 → y=12? Wait, but let's check coordinates. A is (4,2), B is (-1, y). So the run from 4 to -1 is -5, rise is y - 2. Slope is (y - 2)/(-5) = -2 → y - 2 = 10 → y=12. Wait, maybe that's correct. Wait, but let's verify. If y=12, then AB has slope (12-2)/(-1-4)=10/(-5)=-2, which is perpendicular to slope 1/2 (since -2*(1/2)=-1). Yes, that works.

Answer:

\( y = 12 \)