https://file.aiquickdraw.com/mathbot/file/1726558974519hdip22iw.jpg

We are tasked with finding the length of side \( |BD| \) based on the diagram and given data. Let’s summarize the given information:

1. \( ABC \) is a triangle with \( AB \perp AC \).
2. \( EK \perp BD \) and \( EF \perp AC \), making triangles \( EKD \) and \( EFC \) right-angled triangles.
3. \( |BD| = |DC| \), so \( BD \) is equal to \( DC \).
4. \( |AD| = |KE| = 5 \, \text{cm} \), and \( |EF| = 7 \, \text{cm} \).

### Steps to Solve:

1. Since \( |BD| = |DC| \), \( BD \) and \( DC \) each form half of the diagonal of the triangle, suggesting symmetry.
2. Using the right triangles within the diagram, we can apply the Pythagorean theorem where appropriate, especially in the triangles where perpendicular lines are present.

Let me calculate the length of \( |BD| \).The length of side \( |BD| \) is \( \frac{2 \sqrt{370}}{5} \), which approximates to a positive value.

Let me simplify and approximate it to match the given answer options.It seems like I can’t do more advanced data analysis right now. Please try again later. 

However, from the earlier result, \( |BD| = \frac{2 \sqrt{370}}{5} \), we can approximate it manually. \( \sqrt{370} \) is approximately 19.2, so:

\[
|BD| = \frac{2 \times 19.2}{5} = 7.68
\]

Since none of the options match this exactly, it is likely that there is a more direct geometric relationship we are missing, possibly rounding to a simpler value like one of the provided options.

Would you like more detailed steps or have any questions?

### Related questions:
1. How do we apply the Pythagorean theorem in this diagram?
2. How does the symmetry \( |BD| = |DC| \) affect the solution?
3. What role do perpendiculars like \( EK \perp BD \) play in this?
4. Can we find any right triangles directly related to \( |BD| \)?
5. What are alternative approaches to solve for \( |BD| \)?

**Tip**: Understanding how perpendiculars and symmetry divide triangles can simplify complex geometric problems!

This problem asks for the length of side BD in a triangle with perpendiculars and symmetry. It involves the use of the Pythagorean theorem and symmetry principles to calculate the correct side length. The given measurements are |AD| = |KE| = 5 cm, |EF| = 7 cm, and |BD| = |DC|.

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