1. **Problem statement:** We have a right triangle with sides 3, 4, and 5, where $AB=5$, $AC=4$, and $BC=3$. The triangle is folded so that point $A$ falls on point $B$. We need to find the length of the crease formed by this fold. 2. **Understanding the fold:** Folding the triangle so that $A$ lands on $B$ means the crease is the line of reflection that maps $A$ onto $B$. This crease is the perpendicular bisector of segment $AB$. 3. **Find midpoint of $AB$:** The crease passes through the midpoint $M$ of $AB$. Since $AB=5$, the midpoint divides $AB$ into two segments of length $2.5$ each. 4. **Find the crease length:** The crease is the segment inside the triangle along the perpendicular bisector of $AB$. We need to find where this perpendicular bisector intersects the triangle. 5. **Coordinates setup:** Place $C$ at origin $(0,0)$, $A$ at $(0,4)$ (since $AC=4$ vertical), and $B$ at $(3,0)$ (since $BC=3$ horizontal). Then $AB$ connects $(0,4)$ to $(3,0)$. 6. **Midpoint $M$ of $AB$:** $$M=\left(\frac{0+3}{2},\frac{4+0}{2}\right)=(1.5,2)$$ 7. **Slope of $AB$:** $$m_{AB}=\frac{0-4}{3-0}=-\frac{4}{3}$$ 8. **Slope of crease (perpendicular bisector):** $$m_{crease}=\frac{3}{4}$$ 9. **Equation of crease line:** $$y-2=\frac{3}{4}(x-1.5)$$ 10. **Find intersections of crease with sides $AC$ and $BC$:** - Side $AC$ is vertical line $x=0$. - Side $BC$ is horizontal line $y=0$. 11. **Intersection with $AC$ ($x=0$):** $$y-2=\frac{3}{4}(0-1.5)\Rightarrow y-2=-\frac{9}{8}\Rightarrow y=2-1.125=0.875$$ Point is $(0,0.875)$. 12. **Intersection with $BC$ ($y=0$):** $$0-2=\frac{3}{4}(x-1.5)\Rightarrow -2=\frac{3}{4}x-\frac{9}{8}$$ $$\frac{3}{4}x=-2+\frac{9}{8}=-\frac{7}{8}$$ $$x=-\frac{7}{8} \times \frac{4}{3}=-\frac{7}{6} \approx -1.1667$$ This point is outside the triangle since $x$ is negative and $BC$ is between $x=0$ and $x=3$. 13. **Intersection with $AB$ is midpoint $M$, so crease segment inside triangle is between $M$ and intersection with $AC$ at $(0,0.875)$. 14. **Length of crease segment:** $$\text{Length} = \sqrt{(1.5-0)^2 + (2-0.875)^2} = \sqrt{2.25 + 1.265625} = \sqrt{3.515625} \approx 1.875$$ **Final answer:** The length of the crease is approximately $1.875$ units.