https://file.aiquickdraw.com/mathbot/file/1728970299535vud8s02o.jpg

Let's work through the problem in the image. We have a line segment \( AB \), with points \( C \) and \( D \) dividing it, and the lengths are expressed in terms of \( x \) and \( y \):

- \( AC = 2x - 1 \)
- \( CD = 2x - y + 2 \)
- \( BD = x - y + 4 \)

### Part (a): Find the length of \( AD \).

The length of \( AD \) can be calculated by adding the lengths \( AC \) and \( CD \):

\[
AD = AC + CD
\]
\[
AD = (2x - 1) + (2x - y + 2)
\]
\[
AD = 2x - 1 + 2x - y + 2
\]
\[
AD = 4x - y + 1
\]

Thus, the length of \( AD \) is \( \boxed{4x - y + 1} \).

### Part (b): Prove that \( AB = 3AC \) if \( x = 2 \) cm and \( y = 1 \) cm.

To prove this, let's first find the length of \( AB \):

\[
AB = AC + CD + BD
\]
\[
AB = (2x - 1) + (2x - y + 2) + (x - y + 4)
\]
\[
AB = 2x - 1 + 2x - y + 2 + x - y + 4
\]
\[
AB = 5x - 2y + 5
\]

Now, substitute \( x = 2 \) and \( y = 1 \):

\[
AB = 5(2) - 2(1) + 5
\]
\[
AB = 10 - 2 + 5
\]
\[
AB = 13 \text{ cm}
\]

Next, let's calculate \( 3AC \):

\[
AC = 2x - 1
\]
\[
AC = 2(2) - 1 = 4 - 1 = 3 \text{ cm}
\]

\[
3AC = 3 \times 3 = 9 \text{ cm}
\]

We notice that there might be an inconsistency with the values provided because \( AB \) equals \( 13 \) cm, which is not equal to \( 3AC \) (which is \( 9 \) cm). The condition \( AB = 3AC \) does not hold under the given values for \( x \) and \( y \). This may be an error or requires additional information or corrections.

Would you like to go over any part in detail or have further questions?

### Related Questions:
1. What happens if you change the values of \( x \) and \( y \)? Does \( AB = 3AC \) ever hold?
2. Can you verify the expression for \( BD \) and check if there are any constraints missing for \( x \) and \( y \)?
3. How would you solve this problem using coordinate geometry instead of algebraic lengths?
4. If \( AC = 2x - 1 \), what happens when \( x = 0 \)? Is the segment length valid?
5. Can you find the midpoint of \( AB \) using the given expressions?

### Tip:
Always double-check if substituting values into an equation maintains the intended relationships—sometimes it reveals inconsistencies or errors in the given conditions.

Find the length of AD and prove that AB = 3AC if x = 2 cm and y = 1 cm.

Solve a segment addition problem to find the length of AD, and prove that AB = 3AC given values for x and y. This problem involves algebraic expressions for line segments and tests understanding of geometric proofs, suitable for Grades 9-11.

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