Problem Analysis:

We are given a triangle $\triangle ABC$ where:

1. A line $BD$ is drawn such that $\angle ABD = \angle BCD$. This indicates that $BD$ is the angle bisector of $\angle ABC$.
2. On side $AC$, segments $AD$ and $DC$ are marked, with $AD + DC = AC = 10$.
3. The triangle sides $AB$ and $AC$ are given as $AB = 8$ and $AC = 10$.

By the Angle Bisector Theorem, the ratio of the two segments $AD$ and $DC$ on $AC$ is equal to the ratio of the sides $AB$ and $BC$: $\frac{AD}{DC} = \frac{AB}{BC}.$

Step-by-Step Solution:

1. Let $AD = x$ and $DC = 10 - x$: Using the angle bisector theorem: $\frac{AD}{DC} = \frac{AB}{BC}.$ Substituting the lengths: $\frac{x}{10 - x} = \frac{8}{BC}.$

2. Express $BC$ in terms of $x$: Cross-multiplying: $x \cdot BC = 8 \cdot (10 - x).$ Simplify: $BC \cdot x = 80 - 8x.$ $BC = \frac{80}{x + 8}.$

3. Calculate $x$: Since $AD + DC = AC = 10$, substituting values ensures that: $AD + DC = x + (10 - x) = 10.$ Solving: Continuing from where we left off:

Substitute the length expressions directly to solve:

Using $AD = x$ and $DC = 10 - x$, we maintain proportionality. sympy import symbols, Eq, solve

Variables

x = symbols('x') # AD length BC = symbols('BC') # BC length

Given relationships

AB = 8 # Length of AB AC = 10 # Total length AC = AD + DC

Using angle bisector theorem

x / (10 - x) = AB / BC

ratio_eq = Eq(x / (10 - x), AB / BC)

Solve for BC in terms of x

BC_expr = solve(ratio_eq, BC)[0]

Simplify to solve for x based on total AC constraint

BC = 80 / (x + 8)

BC_constraint = Eq(BC_expr, 80 / (x + 8))

Solve for x

AD_solution = solve(BC_constraint, x) AD_solutionThe solution for $x$ (length of $AD$) is:

$x = -4 + 4\sqrt{6} \quad \text{(valid as length cannot be negative)}.$

Now, calculate $DC$: $DC = 10 - x = 10 - (-4 + 4\sqrt{6}) = 14 - 4\sqrt{6}.$

Final Answer:

The length of $DC$ is: $DC = 14 - 4\sqrt{6}.$

Let me know if you want further clarification or approximations!

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Here are some related questions you might find helpful:

1. What is the approximate value of $DC$ in decimal form?
2. Can we determine the length of $BC$?
3. How does the angle bisector theorem help in dividing sides proportionally?
4. What are the possible interpretations if $BD$ wasn't an angle bisector?
5. How can we confirm this calculation geometrically?

Tip: Always check the feasibility of the lengths (e.g., positive values) when solving geometric problems.