We are given a rectangle ABCD with the following details:

• AB = 12 cm (length),
• BC = 9 cm (width),
• A line is drawn through point C, parallel to line BD, and it intersects line AD at point E.

We are tasked with calculating the length of segment CE.

Step-by-step solution:

1. Understand the geometry:

   • ABCD is a rectangle, so $AB \parallel CD$ and $AD \parallel BC$. Additionally, diagonals BD and AC are equal in length and intersect each other at the midpoint, since the diagonals of a rectangle bisect each other.
   • A line is drawn through point $C$, parallel to diagonal $BD$, and intersects $AD$ at point $E$.

2. Slope considerations:

   • Since the line passing through $C$ is parallel to $BD$, it will have the same slope as $BD$.
   • The coordinates of the vertices are as follows (assuming $A$ is the origin):
     • $A(0, 0)$,
     • $B(12, 0)$,
     • $C(12, 9)$,
     • $D(0, 9)$.
   • The slope of diagonal $BD$ is: $m_{BD} = \frac{9 - 0}{0 - 12} = -\frac{3}{4}.$
   • The line through $C(12, 9)$ and parallel to $BD$ will have the same slope, so its equation can be written as: $y - 9 = -\frac{3}{4}(x - 12).$

3. Find the intersection point $E$:

   • The equation of line $AD$ (which is vertical) is $x = 0$, as $A(0, 0)$ and $D(0, 9)$ lie on this vertical line.
   • Substituting $x = 0$ into the equation of the line through $C$: $y - 9 = -\frac{3}{4}(0 - 12) = 9.$ So, $y - 9 = 9$, and therefore $y = 18$.
   • Thus, point $E$ has coordinates $(0, 18)$.

4. Calculate the length of segment $CE$:

   • Using the distance formula between points $C(12, 9)$ and $E(0, 18)$: $CE = \sqrt{(12 - 0)^2 + (9 - 18)^2} = \sqrt{12^2 + (-9)^2} = \sqrt{144 + 81} = \sqrt{225} = 15 \, \text{cm}.$

Thus, the length of segment $CE$ is 15 cm.

Would you like to see further details or explanations?

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Here are 5 related questions:

1. How do you calculate the slope of a line between two points?
2. What are the properties of the diagonals of a rectangle?
3. How do you derive the equation of a line given its slope and a point on the line?
4. How can you verify if two lines are parallel?
5. What is the distance formula, and how does it work for different geometric figures?

Tip: When solving geometry problems, sketching the figure often simplifies the understanding and helps in visualizing relationships between lines and points.