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In the given geometric diagram, we have a triangle \( \triangle ABC \), where \( AD \) is perpendicular to \( BC \), creating a right triangle at \( D \). The expressions for side lengths are given as:

- \( BD = 2x + 10 \)
- \( DA = 3x + 2 \)

We can assume that \( \triangle ABD \) and \( \triangle CAD \) are congruent by the right angle and the corresponding equal sides, implying that \( BD = CD \). Let's solve for \( x \) and the side lengths.

### Step 1: Set up the equation for \( BD = CD \)
Since \( \triangle ABD \cong \triangle ACD \), we have:
\[
BD = CD
\]
Thus:
\[
2x + 10 = 3x + 2
\]

### Step 2: Solve for \( x \)
We subtract \( 2x \) from both sides:
\[
10 = x + 2
\]
Now, subtract 2 from both sides:
\[
x = 8
\]

### Step 3: Find the lengths of \( AD \) and \( CD \)
Now that we have \( x = 8 \), substitute into the expressions for \( AD \) and \( CD \):

- \( AD = 3x + 2 = 3(8) + 2 = 24 + 2 = 26 \)
- \( BD = 2x + 10 = 2(8) + 10 = 16 + 10 = 26 \)

Since \( BD = CD \), we conclude that:
\[
CD = 26
\]

### Final answers:
- \( x = 8 \)
- \( AD = 26 \)
- \( CD = 26 \)

Would you like further details or have any other questions? Here are 5 related questions to explore further:

1. What are the criteria for triangles to be congruent?
2. How do you solve for unknowns in geometric problems with algebraic expressions?
3. What is the Pythagorean theorem, and can it be applied here?
4. How do perpendicular bisectors relate to triangle properties?
5. What are the properties of an isosceles triangle?

**Tip:** In geometry, identifying congruent triangles can often simplify calculations by relating side lengths directly through equality.

Solve for x, AD, and CD in the given geometric diagram where BD = 2x + 10 and AD = 3x + 2, with BD = CD.

Learn how to solve for x, AD, and CD in a geometric problem involving a right triangle. This problem covers algebra and triangle congruence, suitable for students in Grades 8-10.

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