Sure! To find the value of $m$, we apply the angle bisector theorem, which states that the angle bisector divides the opposite side in proportion to the lengths of the other two sides.
Given the lines $y = x$ and $y = mx$, the slope of the line $y = x$ is 1, and the slope of the line $y = mx$ is $m$.
The angle bisector has a slope that is the geometric mean of the slopes of the lines that form the angle. The geometric mean of two numbers $a$ and $b$ is $\sqrt{ab}$.
So, the geometric mean of the slopes of the lines $y = x$ and $y = mx$ is $\sqrt{1 \cdot m} = \sqrt{m}$.
Now, we know that the line $y = \frac{1}{7}x$ bisects the angle between these two lines. Therefore, the slope of the line $y = \frac{1}{7}x$ is equal to the geometric mean of the slopes, which is $\sqrt{m}$.
Since the slope of the line $y = \frac{1}{7}x$ is $\frac{1}{7}$, we can solve for $m$ by setting $\sqrt{m}$ equal to $\frac{1}{7}$ and then squaring both sides to solve for $m$.
So, we have $\sqrt{m} = \frac{1}{7}$, and squaring both sides gives $m = \left(\frac{1}{7}\right)^2 = \frac{1}{49}$.
Therefore, the value of $m$ is $\frac{1}{49}$.
So, I made a mistake in my previous response, and I appreciate your patience. The correct value of $m$ is $\frac{1}{49}$, not $\frac{\sqrt{7}}{7}$.