https://file.aiquickdraw.com/mathbot/file/1726104142182pyij9nf3.jpg

The image shows two intersecting lines forming a set of congruent segments on the lines, marked by tick marks. The lines intersect at point \( D \), and we are given algebraic expressions for line segments \( DC \) and \( DA \):

- \( DC = 6x \)
- \( DA = 4x + 18 \)

From the markings, we can see that \( DC \) and \( DA \) are equal in length, meaning:

\[
DC = DA
\]

Thus, we can set the equations equal to each other:

\[
6x = 4x + 18
\]

Now, let's solve for \( x \).

\[
6x - 4x = 18
\]
\[
2x = 18
\]
\[
x = 9
\]

Now that we know \( x = 9 \), we can substitute it back into the expressions for \( DC \) and \( DA \):

\[
DC = 6x = 6(9) = 54
\]
\[
DA = 4x + 18 = 4(9) + 18 = 36 + 18 = 54
\]

Thus, both segments \( DC \) and \( DA \) are 54 units long, confirming the congruence.

Would you like further details or have any other questions?

### 5 Related Questions:
1. How would the solution change if \( DA = 5x + 12 \) instead of \( 4x + 18 \)?
2. What is the significance of the tick marks on the lines in geometric terms?
3. Can we apply the same method if we are given \( DE = 8x \) and \( DB = 10x + 5 \)?
4. How do we solve similar problems involving systems of equations?
5. What if the lines were not intersecting at a right angle—how would the solution be affected?

### Tip:
When two segments are marked congruent in geometry, use algebra to express their equality to find unknown variables.

Given two intersecting lines forming congruent segments, where DC = 6x and DA = 4x + 18, solve for x and find the length of DC and DA.

Learn how to solve for x and find the lengths of the congruent line segments DC and DA. This problem involves basic algebra and geometry, using the relationship between congruent segments. Suitable for students in Grades 7-9.

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