Let's break down the problem step by step.

Step 1: Understanding the Problem

We know that:

• Point B is between A and C on line segment $AC$.
• $AB = 13 + 2x$
• $BC = 12$
• $AC = x + 32$

Since $B$ is between $A$ and $C$, we can say: $AB + BC = AC$

Step 2: Write the Equation

Using the given expressions for $AB$, $BC$, and $AC$, we can set up the equation: $(13 + 2x) + 12 = x + 32$

Step 3: Simplify and Solve for $x$

Now, let's solve this equation:

$13 + 2x + 12 = x + 32$ $2x + 25 = x + 32$

Subtract $x$ from both sides:

$x + 25 = 32$

Subtract 25 from both sides:

$x = 7$

Step 4: Find AB, BC, and AC

Now that we have $x = 7$, we can find the lengths of $AB$, $BC$, and $AC$:

• $AB = 13 + 2x = 13 + 2(7) = 13 + 14 = 27$
• $BC = 12$ (given)
• $AC = x + 32 = 7 + 32 = 39$

Final Answers:

• $AB = 27$
• $BC = 12$
• $AC = 39$

Do you want details or have any questions?

Here are 5 related questions to expand on this topic:

1. How do you determine the position of point B on line segment AC?
2. What happens to the lengths of AB, BC, and AC if x increases or decreases?
3. Can you write an equation if AB, BC, and AC are given as expressions in terms of different variables?
4. How can this type of equation be applied in real-world scenarios involving distances?
5. How would the problem change if B were not directly between A and C?

Tip: Always check if the calculated lengths add up correctly to the total length to verify your solution.