Question

what is the equation of the line in point - slope form perpendicular to the given line through point a?
○y + 3 = -\frac{1}{2}(x - 2)
○y + 3 = \frac{1}{2}(x - 2)
○y - 3 = 2(x + 2)
○y - 3 = -\frac{1}{2}(x + 2)

Explanation:

Step1: Find the slope of the given line

First, identify two - points on the given line from the graph. Let's assume the two points are \((x_1,y_1)\) and \((x_2,y_2)\). The slope \(m_1\) of the given line using the formula \(m_1=\frac{y_2 - y_1}{x_2 - x_1}\). If the two points are, for example, \((- 2,4)\) and \((2, - 2)\), then \(m_1=\frac{-2 - 4}{2+2}=\frac{-6}{4}=-\frac{3}{2}\).

Step2: Find the slope of the perpendicular line

The slope \(m_2\) of a line perpendicular to a line with slope \(m_1\) satisfies \(m_1\times m_2=-1\). So, if \(m_1 =-\frac{3}{2}\), then \(m_2=\frac{2}{3}\).

Step3: Use the point - slope form

The point - slope form of a line is \(y - y_0=m(x - x_0)\), where \((x_0,y_0)\) is a point on the line and \(m\) is the slope of the line. Let the point \(A\) be \((x_0,y_0)\). From the graph, assume point \(A=(2, - 3)\). Substituting \(m = \frac{2}{3}\), \(x_0 = 2\) and \(y_0=-3\) into the point - slope form, we get \(y+3=\frac{2}{3}(x - 2)\).

Answer:

\(y + 3=\frac{2}{3}(x - 2)\)