Let's analyze the problem given in the image.

The information provided:

• $AC = 24$
• $AB = 6x - 6$
• $BC = 5x - 3$
• $BE = 3x + 2$

The questions ask which of the following statements are true:

1. $AC$ bisects $DE$.
2. $DE$ bisects $AC$.
3. $DE$ bisects $\angle AC$.

We need to analyze these using the values of $x$ and given equations. Let's first solve for $x$.

Step 1: Finding $x$

From the figure, we know that: $AC = AB + BC$ Thus: $24 = (6x - 6) + (5x - 3)$ Simplify: $24 = 6x - 6 + 5x - 3$ $24 = 11x - 9$ $11x = 33$ $x = 3$

Step 2: Substituting $x$ to find segment lengths

Now, substitute $x = 3$ into the expressions for $AB$, $BC$, and $BE$.

• $AB = 6(3) - 6 = 18 - 6 = 12$
• $BC = 5(3) - 3 = 15 - 3 = 12$
• $BE = 3(3) + 2 = 9 + 2 = 11$

Step 3: Verifying the statements

1. Does $AC$ bisect $DE$?

   • If $AC$ bisects $DE$, then $AD = AE$, but there is no information given in the problem to suggest that this is true. Therefore, Statement I is false.

2. Does $DE$ bisect $AC$?

   • For $DE$ to bisect $AC$, point $E$ must be the midpoint of $AC$. Since $AB = BC = 12$, point $E$ is the midpoint. Therefore, Statement II is true.

3. Does $DE$ bisect $\angle AC$?

   • We don't have any information about the angles in this setup, so there is no way to conclude whether $DE$ bisects $\angle AC$. Therefore, Statement III is false.

Final Answer:

The correct answer is b) II only.

Do you want more details or have any questions?

Follow-up Questions:

1. How would the problem change if $AB$ and $BC$ were not equal?
2. What is the significance of the bisector in geometry problems?
3. Can we find the lengths of other segments in the figure based on the given information?
4. What additional information would you need to verify Statement I?
5. How does solving for $x$ help in geometry problems?

Tip:

Always start with finding segment lengths or relationships in geometry questions when solving for bisectors or midpoints.