Answer
$x = 2.5$
Work Step by Step
According to the triangle-angle-bisector theorem, when a ray bisects the angle of a triangle, the opposite side is divided into two segments that are proportional to the remaining two sides of the triangle.
Let's set up a proportion that compares the segments of the intersected side to the other two sides:
$\frac{5x}{6x} = \frac{7x}{10x - 4}$
Use the cross product property to get rid of the fractions:
$5x(10x - 4) = 6x(7x)$
Use the distributive property:
$50x^2 - 20x = 42x^2$
Subtract $42x^2$ from each side to move all terms to the left side of the equation:
$50x^2 - 42x^2 - 20x = 0$
Combine like terms:
$8x^2 - 20x = 0$
Factor out $4$ from each term:
$2x^2 - 5x = 0$
Factor out $x$ from the left side of the equation:
$x(2x - 5) = 0$
Set each factor equal to $0$:
$x = 0$ or $2x - 5 = 0$
We can discard $x = 0$ because we cannot have lengths equal to $0$.
Let's look at the other factor:
$2x - 5 = 0$
Add $5$ to each side of the equation:
$2x = 5$
Divide each side of the equation by $2$ to solve for $x$:
$x = 2.5$
