abcd is a rectangle. ab equals 12 cm, bc equals 9 cm. the parallel line thru the point c at the bd line intersecting ad in point e. calculate the lenght of the segment ce

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.

abcd is a rectangle. ab equals 12 cm, bc equals 9 cm. the parallel line thru the point c at the bd line intersecting ad in point e. calculate the length of the segment ce.

Calculate the length of segment CE in a rectangle ABCD, where AB = 12 cm and BC = 9 cm, with a parallel line through point C and intersecting line AD at point E. This problem involves using the slope and distance formulas, and it is suitable for students in Grades 9-10.

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