Answer
$x = 4$
Work Step by Step
$R$ is the incenter of the triangle because it marks the point of concurrency of a triangle's angle bisectors. Therefore, $R$ is the center of the circle that is inscribed inside the triangle, so any point on that circle is equidistant from $R$.
In this diagram, $S$ and $T$ are located on the inscribed circle; therefore, $RS$ is equal to $RT$. We can now set these distances equal to one another to solve for $x$:
$4(x - 3) + 6 = 5(2x - 6)$
Let's distribute first to simplify:
$4x - 12 + 6 = 10x - 30$
Add the constants on the left side of the equation:
$4x - 6 = 10x - 30$
Subtract $4x$ from each side of the equation to isolate the variable on one side of the equation:
$-6 = 6x - 30$
Add $30$ to each side of the equation to isolate constants on one side of the equation:
$6x = 24$
Divide each side of the equation by $6$ to solve for $x$:
$x = 4$
