Line through the points (-2, 6) and (4, 8) is perpendicular to the line through the points (8, 12) and (x, 24). Find the value of x.

Step 1: Calculate the slope of the first line

To find the slope of the first line, we use the given points $(-2, 6)$ and $(4, 8)$. The formula for the slope $m$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is $m = \frac{y_2 - y_1}{x_2 - x_1}$. Substituting the coordinates, we get the slope of the first line.

Step 2: Calculate the slope of the second line

Similarly, we calculate the slope of the second line using the points $(8, 12)$ and $(x, 24)$. We apply the same slope formula to these points. This will give us an expression for the slope $m_2$ in terms of $x$.

Step 3: Apply the condition for perpendicular lines

Two lines are perpendicular if the product of their slopes is $-1$. We have calculated $m_1 = \frac{1}{3}$ and $m_2 = \frac{12}{x-8}$. We set their product equal to $-1$ to form an equation that we can solve for $x$.

Step 4: Solve for x

Now we solve the equation for $x$. We simplify the product of the slopes, then multiply both sides by $(x-8)$, and finally isolate $x$ to find its value. This gives us the required coordinate for the second line.

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