1. **State the problem:** We have a line given by the equation $y = x + 4$ and it intersects the x-axis at point $C$. We want to find the lengths of line segments $AC$ and $BC$. However, points $A$ and $B$ are not defined in the problem, so we will assume $A$ and $B$ are points on the line or axes related to the intersection. 2. **Find point $C$:** The x-axis is defined by $y=0$. To find $C$, set $y=0$ in the line equation: $$0 = x + 4$$ $$x = -4$$ So, $C = (-4, 0)$. 3. **Assuming points $A$ and $B$:** Since the problem mentions the Pythagorean theorem and segments $AC$ and $BC$, a common interpretation is that $A$ and $B$ are the points where the line intersects the y-axis and the origin respectively. - Point $A$ is the y-intercept of the line $y = x + 4$, which occurs when $x=0$: $$y = 0 + 4 = 4$$ So, $A = (0, 4)$. - Point $B$ is the origin $(0,0)$. 4. **Calculate length $AC$:** Use the distance formula between points $A(0,4)$ and $C(-4,0)$: $$AC = \sqrt{(0 - (-4))^2 + (4 - 0)^2} = \sqrt{4^2 + 4^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}$$ 5. **Calculate length $BC$:** Use the distance formula between points $B(0,0)$ and $C(-4,0)$: $$BC = \sqrt{(0 - (-4))^2 + (0 - 0)^2} = \sqrt{4^2 + 0} = 4$$ **Final answer:** - Length of $AC$ is $4\sqrt{2}$. - Length of $BC$ is $4$.