Question

Line ST and point V are shown on the graph.

On a coordinate plane, line S T goes through (negative 5, 0) and (5, 2). Point V is at (0, negative 2).

Line VW is to be drawn on the graph such that it is perpendicular to line ST. If the coordinates of point W are (−1, y), what is the value of y?

−7
−5
2
3

Answer 1

First, we need to find the slope of line ST, which passes through the points \((-5, 0)\) and \((5, 2)\).

The slope \(m\) of a line that passes through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Substituting the coordinates of the given points:

\[
m_{ST} = \frac{2 - 0}{5 - (-5)} = \frac{2}{5 + 5} = \frac{2}{10} = \frac{1}{5}
\]

The slope of line ST is \(\frac{1}{5}\).

Since line VW is perpendicular to line ST, the slope of line VW, denoted as \(m_{VW}\), is the negative reciprocal of the slope of line ST. Therefore:

\[
m_{VW} = -\frac{1}{m_{ST}} = -\frac{1}{\frac{1}{5}} = -5
\]

Next, we know that point V is at \((0, -2)\) and point W is \((-1, y)\). We can use the slope formula again to find \(y\):

\[
m_{VW} = \frac{y - (-2)}{-1 - 0} = \frac{y + 2}{-1} = -(y + 2)
\]

Since the slope \(m_{VW} = -5\), we can set up the equation:

\[
-(y + 2) = -5
\]

Multiplying both sides of the equation by \(-1\):

\[
y + 2 = 5
\]

Now, subtracting 2 from both sides:

\[
y = 5 - 2 = 3
\]

Thus, the value of \(y\) is \(\boxed{3}\).