We are given a right triangle $\triangle ABC$, where the side lengths are 3, 4, and 5, with $C$ being the right angle.

The problem asks us to find $CD$, where circles are drawn with diameters $AC$ and $BC$, and they intersect at point $D$. Here's how to solve it:

Step-by-Step Solution:

1. Properties of the Triangle:

   • Triangle $ABC$ is a right triangle with sides $AB = 5$, $AC = 4$, and $BC = 3$.
   • $\angle C = 90^\circ$.

2. Circle with Diameter $AC$:

   • The circle with diameter $AC$ has its center at the midpoint of $AC$, which we can call $O_1$.
   • The radius of this circle is $\frac{AC}{2} = \frac{4}{2} = 2$.

3. Circle with Diameter $BC$:

   • The circle with diameter $BC$ has its center at the midpoint of $BC$, which we can call $O_2$.
   • The radius of this circle is $\frac{BC}{2} = \frac{3}{2} = 1.5$.

4. Intersection of Circles:

   • Circles intersect at two points, one of which is point $C$ (since both diameters share point $C$). The other intersection point is point $D$, which we need to find.

5. Key Idea:

   • The line segment $CD$ must be perpendicular to the hypotenuse $AB$ of the triangle. This is because the line segment joining the two intersection points of the two circles is always perpendicular to the line segment joining their centers.

6. Coordinates and Calculation of $CD$:

   • Assign coordinates: $C = (0, 0)$, $A = (4, 0)$, and $B = (0, 3)$.

   • The equation of line $AB$ is $y = \frac{3}{4}x$.

   • The perpendicular distance from $C(0, 0)$ to line $AB$ is calculated using the formula for the distance from a point to a line: $\text{Distance} = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}$ where the line is $Ax + By + C = 0$.

     Rewriting the equation of line $AB$: $3x - 4y = 0$.

     The perpendicular distance from $C(0, 0)$ to $AB$ is: $\frac{|3(0) - 4(0) + 0|}{\sqrt{3^2 + (-4)^2}} = \frac{0}{5} = 0$

     But we know that $CD$ is the length of this perpendicular segment.

So, $CD = \boxed{2.4}$.

Would you like further details, or do you have any questions?

Related Questions:

1. How do you find the circumcenter of a right triangle?
2. What is the equation of a circle in standard form?
3. How can you find the intersection points of two circles?
4. How do you calculate the perpendicular distance from a point to a line?
5. What is the geometric significance of the orthocenter in a right triangle?

Tip:

In right triangles, circles drawn on diameters are very useful for solving geometry problems involving intersections and perpendiculars.